P. 501 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 50001-50100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	swapf2f1oaALT 50001	Alternate proof of swapf2f1oa 50000. (Contributed by Zhi Wang, 8-Oct-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨𝑂, 𝑃⟩)    &   ⊢ 𝑆 = (𝐶 ×c 𝐷)    &   ⊢ 𝑇 = (𝐷 ×c 𝐶)    &   ⊢ 𝐻 = (Hom ‘𝑆)    &   ⊢ 𝐽 = (Hom ‘𝑇)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑋𝑃𝑌):(𝑋𝐻𝑌)–1-1-onto→((𝑂‘𝑋)𝐽(𝑂‘𝑌)))
Theorem	swapfid 50002	Each identity morphism in the source category is mapped to the corresponding identity morphism in the target category. See also swapfida 50003. (Contributed by Zhi Wang, 8-Oct-2025.)
⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ 𝑆 = (𝐶 ×c 𝐷)    &   ⊢ 𝑇 = (𝐷 ×c 𝐶)    &   ⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨𝑂, 𝑃⟩)    &   ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))    &   ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐷))    &   ⊢ 1 = (Id‘𝑆)    &   ⊢ 𝐼 = (Id‘𝑇)    ⇒   ⊢ (𝜑 → ((⟨𝑋, 𝑌⟩𝑃⟨𝑋, 𝑌⟩)‘( 1 ‘⟨𝑋, 𝑌⟩)) = (𝐼‘(𝑂‘⟨𝑋, 𝑌⟩)))
Theorem	swapfida 50003	Each identity morphism in the source category is mapped to the corresponding identity morphism in the target category. See also swapfid 50002. (Contributed by Zhi Wang, 8-Oct-2025.)
⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ 𝑆 = (𝐶 ×c 𝐷)    &   ⊢ 𝑇 = (𝐷 ×c 𝐶)    &   ⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨𝑂, 𝑃⟩)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ 1 = (Id‘𝑆)    &   ⊢ 𝐼 = (Id‘𝑇)    ⇒   ⊢ (𝜑 → ((𝑋𝑃𝑋)‘( 1 ‘𝑋)) = (𝐼‘(𝑂‘𝑋)))
Theorem	swapfcoa 50004	Composition in the source category is mapped to composition in the target. (𝜑 → 𝐶 ∈ Cat) and (𝜑 → 𝐷 ∈ Cat) can be replaced by a weaker hypothesis (𝜑 → 𝑆 ∈ Cat). (Contributed by Zhi Wang, 8-Oct-2025.)
⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ 𝑆 = (𝐶 ×c 𝐷)    &   ⊢ 𝑇 = (𝐷 ×c 𝐶)    &   ⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨𝑂, 𝑃⟩)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ 𝐻 = (Hom ‘𝑆)    &   ⊢ (𝜑 → 𝑀 ∈ (𝑋𝐻𝑌))    &   ⊢ (𝜑 → 𝑁 ∈ (𝑌𝐻𝑍))    &   ⊢ · = (comp‘𝑆)    &   ⊢ ∙ = (comp‘𝑇)    ⇒   ⊢ (𝜑 → ((𝑋𝑃𝑍)‘(𝑁(⟨𝑋, 𝑌⟩ · 𝑍)𝑀)) = (((𝑌𝑃𝑍)‘𝑁)(⟨(𝑂‘𝑋), (𝑂‘𝑌)⟩ ∙ (𝑂‘𝑍))((𝑋𝑃𝑌)‘𝑀)))
Theorem	swapffunc 50005	The swap functor is a functor. (Contributed by Zhi Wang, 8-Oct-2025.)
⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ 𝑆 = (𝐶 ×c 𝐷)    &   ⊢ 𝑇 = (𝐷 ×c 𝐶)    &   ⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨𝑂, 𝑃⟩)    ⇒   ⊢ (𝜑 → 𝑂(𝑆 Func 𝑇)𝑃)
Theorem	swapfffth 50006	The swap functor is a fully faithful functor. (Contributed by Zhi Wang, 8-Oct-2025.)
⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ 𝑆 = (𝐶 ×c 𝐷)    &   ⊢ 𝑇 = (𝐷 ×c 𝐶)    &   ⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨𝑂, 𝑃⟩)    ⇒   ⊢ (𝜑 → 𝑂((𝑆 Full 𝑇) ∩ (𝑆 Faith 𝑇))𝑃)
Theorem	swapffunca 50007	The swap functor is a functor. (Contributed by Zhi Wang, 9-Oct-2025.)
⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ 𝑆 = (𝐶 ×c 𝐷)    &   ⊢ 𝑇 = (𝐷 ×c 𝐶)    ⇒   ⊢ (𝜑 → (𝐶 swapF 𝐷) ∈ (𝑆 Func 𝑇))
Theorem	swapfiso 50008	The swap functor is an isomorphism between product categories. (Contributed by Zhi Wang, 8-Oct-2025.)
⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ 𝑆 = (𝐶 ×c 𝐷)    &   ⊢ 𝑇 = (𝐷 ×c 𝐶)    &   ⊢ 𝐸 = (CatCat‘𝑈)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑈)    &   ⊢ (𝜑 → 𝑇 ∈ 𝑈)    &   ⊢ 𝐼 = (Iso‘𝐸)    ⇒   ⊢ (𝜑 → (𝐶 swapF 𝐷) ∈ (𝑆𝐼𝑇))
Theorem	swapciso 50009	The product category is categorically isomorphic to the swapped product category. (Contributed by Zhi Wang, 8-Oct-2025.)
⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ 𝑆 = (𝐶 ×c 𝐷)    &   ⊢ 𝑇 = (𝐷 ×c 𝐶)    &   ⊢ 𝐸 = (CatCat‘𝑈)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑈)    &   ⊢ (𝜑 → 𝑇 ∈ 𝑈)    ⇒   ⊢ (𝜑 → 𝑆( ≃𝑐 ‘𝐸)𝑇)
21.50.15.15  Functor evaluation
Theorem	oppc1stflem 50010*	A utility theorem for proving theorems on projection functors of opposite categories. (Contributed by Zhi Wang, 19-Nov-2025.)
⊢ 𝑂 = (oppCat‘𝐶)    &   ⊢ 𝑃 = (oppCat‘𝐷)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑊)    &   ⊢ ((𝜑 ∧ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) → ( oppFunc ‘(𝐶𝐹𝐷)) = (𝑂𝐹𝑃))    &   ⊢ 𝐹 = (𝑐 ∈ Cat, 𝑑 ∈ Cat ↦ 𝑌)    ⇒   ⊢ (𝜑 → ( oppFunc ‘(𝐶𝐹𝐷)) = (𝑂𝐹𝑃))
Theorem	oppc1stf 50011	The opposite functor of the first projection functor is the first projection functor of opposite categories. (Contributed by Zhi Wang, 19-Nov-2025.)
⊢ 𝑂 = (oppCat‘𝐶)    &   ⊢ 𝑃 = (oppCat‘𝐷)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑊)    ⇒   ⊢ (𝜑 → ( oppFunc ‘(𝐶 1stF 𝐷)) = (𝑂 1stF 𝑃))
Theorem	oppc2ndf 50012	The opposite functor of the second projection functor is the second projection functor of opposite categories. (Contributed by Zhi Wang, 19-Nov-2025.)
⊢ 𝑂 = (oppCat‘𝐶)    &   ⊢ 𝑃 = (oppCat‘𝐷)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑊)    ⇒   ⊢ (𝜑 → ( oppFunc ‘(𝐶 2ndF 𝐷)) = (𝑂 2ndF 𝑃))
Theorem	1stfpropd 50013	If two categories have the same set of objects, morphisms, and compositions, then they have same first projection functors. (Contributed by Zhi Wang, 20-Nov-2025.)
⊢ (𝜑 → (Homf ‘𝐴) = (Homf ‘𝐵))    &   ⊢ (𝜑 → (compf‘𝐴) = (compf‘𝐵))    &   ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))    &   ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷))    &   ⊢ (𝜑 → 𝐴 ∈ Cat)    &   ⊢ (𝜑 → 𝐵 ∈ Cat)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    ⇒   ⊢ (𝜑 → (𝐴 1stF 𝐶) = (𝐵 1stF 𝐷))
Theorem	2ndfpropd 50014	If two categories have the same set of objects, morphisms, and compositions, then they have same second projection functors. (Contributed by Zhi Wang, 20-Nov-2025.)
⊢ (𝜑 → (Homf ‘𝐴) = (Homf ‘𝐵))    &   ⊢ (𝜑 → (compf‘𝐴) = (compf‘𝐵))    &   ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))    &   ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷))    &   ⊢ (𝜑 → 𝐴 ∈ Cat)    &   ⊢ (𝜑 → 𝐵 ∈ Cat)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    ⇒   ⊢ (𝜑 → (𝐴 2ndF 𝐶) = (𝐵 2ndF 𝐷))
Theorem	diagpropd 50015	If two categories have the same set of objects, morphisms, and compositions, then they have same diagonal functors. (Contributed by Zhi Wang, 20-Nov-2025.)
⊢ (𝜑 → (Homf ‘𝐴) = (Homf ‘𝐵))    &   ⊢ (𝜑 → (compf‘𝐴) = (compf‘𝐵))    &   ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))    &   ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷))    &   ⊢ (𝜑 → 𝐴 ∈ Cat)    &   ⊢ (𝜑 → 𝐵 ∈ Cat)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    ⇒   ⊢ (𝜑 → (𝐴Δfunc𝐶) = (𝐵Δfunc𝐷))
21.50.15.16  Transposed curry functors
Theorem	cofuswapfcl 50016	The bifunctor pre-composed with a swap functor is a bifunctor. (Contributed by Zhi Wang, 10-Oct-2025.)
⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ (𝜑 → 𝐹 ∈ ((𝐷 ×c 𝐶) Func 𝐸))    &   ⊢ (𝜑 → 𝐺 = (𝐹 ∘func (𝐶 swapF 𝐷)))    ⇒   ⊢ (𝜑 → 𝐺 ∈ ((𝐶 ×c 𝐷) Func 𝐸))
Theorem	cofuswapf1 50017	The object part of a bifunctor pre-composed with a swap functor. (Contributed by Zhi Wang, 9-Oct-2025.)
⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ (𝜑 → 𝐹 ∈ ((𝐷 ×c 𝐶) Func 𝐸))    &   ⊢ (𝜑 → 𝐺 = (𝐹 ∘func (𝐶 swapF 𝐷)))    &   ⊢ 𝐴 = (Base‘𝐶)    &   ⊢ 𝐵 = (Base‘𝐷)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑋(1st ‘𝐺)𝑌) = (𝑌(1st ‘𝐹)𝑋))
Theorem	cofuswapf2 50018	The morphism part of a bifunctor pre-composed with a swap functor. (Contributed by Zhi Wang, 9-Oct-2025.)
⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ (𝜑 → 𝐹 ∈ ((𝐷 ×c 𝐶) Func 𝐸))    &   ⊢ (𝜑 → 𝐺 = (𝐹 ∘func (𝐶 swapF 𝐷)))    &   ⊢ 𝐴 = (Base‘𝐶)    &   ⊢ 𝐵 = (Base‘𝐷)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑊 ∈ 𝐵)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ 𝐽 = (Hom ‘𝐷)    &   ⊢ (𝜑 → 𝑀 ∈ (𝑋𝐻𝑍))    &   ⊢ (𝜑 → 𝑁 ∈ (𝑌𝐽𝑊))    ⇒   ⊢ (𝜑 → (𝑀(⟨𝑋, 𝑌⟩(2nd ‘𝐺)⟨𝑍, 𝑊⟩)𝑁) = (𝑁(⟨𝑌, 𝑋⟩(2nd ‘𝐹)⟨𝑊, 𝑍⟩)𝑀))
Theorem	tposcurf1cl 50019	The partially evaluated transposed curry functor is a functor. (Contributed by Zhi Wang, 9-Oct-2025.)
⊢ (𝜑 → 𝐺 = (⟨𝐶, 𝐷⟩ curryF (𝐹 ∘func (𝐶 swapF 𝐷))))    &   ⊢ 𝐴 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ (𝜑 → 𝐹 ∈ ((𝐷 ×c 𝐶) Func 𝐸))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ (𝜑 → 𝐾 = ((1st ‘𝐺)‘𝑋))    ⇒   ⊢ (𝜑 → 𝐾 ∈ (𝐷 Func 𝐸))
Theorem	tposcurf11 50020	Value of the double evaluated transposed curry functor. (Contributed by Zhi Wang, 9-Oct-2025.)
⊢ (𝜑 → 𝐺 = (⟨𝐶, 𝐷⟩ curryF (𝐹 ∘func (𝐶 swapF 𝐷))))    &   ⊢ 𝐴 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ (𝜑 → 𝐹 ∈ ((𝐷 ×c 𝐶) Func 𝐸))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ (𝜑 → 𝐾 = ((1st ‘𝐺)‘𝑋))    &   ⊢ 𝐵 = (Base‘𝐷)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ((1st ‘𝐾)‘𝑌) = (𝑌(1st ‘𝐹)𝑋))
Theorem	tposcurf12 50021	The partially evaluated transposed curry functor at a morphism. (Contributed by Zhi Wang, 9-Oct-2025.)
⊢ (𝜑 → 𝐺 = (⟨𝐶, 𝐷⟩ curryF (𝐹 ∘func (𝐶 swapF 𝐷))))    &   ⊢ 𝐴 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ (𝜑 → 𝐹 ∈ ((𝐷 ×c 𝐶) Func 𝐸))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ (𝜑 → 𝐾 = ((1st ‘𝐺)‘𝑋))    &   ⊢ 𝐵 = (Base‘𝐷)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ 𝐽 = (Hom ‘𝐷)    &   ⊢ 1 = (Id‘𝐶)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐻 ∈ (𝑌𝐽𝑍))    ⇒   ⊢ (𝜑 → ((𝑌(2nd ‘𝐾)𝑍)‘𝐻) = (𝐻(⟨𝑌, 𝑋⟩(2nd ‘𝐹)⟨𝑍, 𝑋⟩)( 1 ‘𝑋)))
Theorem	tposcurf1 50022*	Value of the object part of the transposed curry functor. (Contributed by Zhi Wang, 9-Oct-2025.)
⊢ (𝜑 → 𝐺 = (⟨𝐶, 𝐷⟩ curryF (𝐹 ∘func (𝐶 swapF 𝐷))))    &   ⊢ 𝐴 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ (𝜑 → 𝐹 ∈ ((𝐷 ×c 𝐶) Func 𝐸))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ (𝜑 → 𝐾 = ((1st ‘𝐺)‘𝑋))    &   ⊢ 𝐵 = (Base‘𝐷)    &   ⊢ 𝐽 = (Hom ‘𝐷)    &   ⊢ 1 = (Id‘𝐶)    ⇒   ⊢ (𝜑 → 𝐾 = ⟨(𝑦 ∈ 𝐵 ↦ (𝑦(1st ‘𝐹)𝑋)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (𝑔(⟨𝑦, 𝑋⟩(2nd ‘𝐹)⟨𝑧, 𝑋⟩)( 1 ‘𝑋))))⟩)
Theorem	tposcurf2 50023*	Value of the transposed curry functor at a morphism. (Contributed by Zhi Wang, 10-Oct-2025.)
⊢ (𝜑 → 𝐺 = (⟨𝐶, 𝐷⟩ curryF (𝐹 ∘func (𝐶 swapF 𝐷))))    &   ⊢ 𝐴 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ (𝜑 → 𝐹 ∈ ((𝐷 ×c 𝐶) Func 𝐸))    &   ⊢ 𝐵 = (Base‘𝐷)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ 𝐼 = (Id‘𝐷)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐴)    &   ⊢ (𝜑 → 𝐾 ∈ (𝑋𝐻𝑌))    &   ⊢ (𝜑 → 𝐿 = ((𝑋(2nd ‘𝐺)𝑌)‘𝐾))    ⇒   ⊢ (𝜑 → 𝐿 = (𝑧 ∈ 𝐵 ↦ ((𝐼‘𝑧)(⟨𝑧, 𝑋⟩(2nd ‘𝐹)⟨𝑧, 𝑌⟩)𝐾)))
Theorem	tposcurf2val 50024	Value of a component of the transposed curry functor natural transformation. (Contributed by Zhi Wang, 10-Oct-2025.)
⊢ (𝜑 → 𝐺 = (⟨𝐶, 𝐷⟩ curryF (𝐹 ∘func (𝐶 swapF 𝐷))))    &   ⊢ 𝐴 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ (𝜑 → 𝐹 ∈ ((𝐷 ×c 𝐶) Func 𝐸))    &   ⊢ 𝐵 = (Base‘𝐷)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ 𝐼 = (Id‘𝐷)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐴)    &   ⊢ (𝜑 → 𝐾 ∈ (𝑋𝐻𝑌))    &   ⊢ (𝜑 → 𝐿 = ((𝑋(2nd ‘𝐺)𝑌)‘𝐾))    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐿‘𝑍) = ((𝐼‘𝑍)(⟨𝑍, 𝑋⟩(2nd ‘𝐹)⟨𝑍, 𝑌⟩)𝐾))
Theorem	tposcurf2cl 50025	The transposed curry functor at a morphism is a natural transformation. (Contributed by Zhi Wang, 10-Oct-2025.)
⊢ (𝜑 → 𝐺 = (⟨𝐶, 𝐷⟩ curryF (𝐹 ∘func (𝐶 swapF 𝐷))))    &   ⊢ 𝐴 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ (𝜑 → 𝐹 ∈ ((𝐷 ×c 𝐶) Func 𝐸))    &   ⊢ 𝐵 = (Base‘𝐷)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ 𝐼 = (Id‘𝐷)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐴)    &   ⊢ (𝜑 → 𝐾 ∈ (𝑋𝐻𝑌))    &   ⊢ (𝜑 → 𝐿 = ((𝑋(2nd ‘𝐺)𝑌)‘𝐾))    &   ⊢ 𝑁 = (𝐷 Nat 𝐸)    ⇒   ⊢ (𝜑 → 𝐿 ∈ (((1st ‘𝐺)‘𝑋)𝑁((1st ‘𝐺)‘𝑌)))
Theorem	tposcurfcl 50026	The transposed curry functor of a functor 𝐹:𝐷 × 𝐶⟶𝐸 is a functor tposcurry (𝐹):𝐶⟶(𝐷⟶𝐸). (Contributed by Zhi Wang, 9-Oct-2025.)
⊢ (𝜑 → 𝐺 = (⟨𝐶, 𝐷⟩ curryF (𝐹 ∘func (𝐶 swapF 𝐷))))    &   ⊢ 𝑄 = (𝐷 FuncCat 𝐸)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ (𝜑 → 𝐹 ∈ ((𝐷 ×c 𝐶) Func 𝐸))    ⇒   ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝑄))
21.50.15.17  Constant functors
Theorem	diag1 50027*	The constant functor of 𝑋. Example 3.20(2) of [Adamek] p. 30. (Contributed by Zhi Wang, 17-Oct-2025.)
⊢ 𝐿 = (𝐶Δfunc𝐷)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ 𝐴 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ 𝐾 = ((1st ‘𝐿)‘𝑋)    &   ⊢ 𝐵 = (Base‘𝐷)    &   ⊢ 𝐽 = (Hom ‘𝐷)    &   ⊢ 1 = (Id‘𝐶)    ⇒   ⊢ (𝜑 → 𝐾 = ⟨(𝑦 ∈ 𝐵 ↦ 𝑋), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑓 ∈ (𝑦𝐽𝑧) ↦ ( 1 ‘𝑋)))⟩)
Theorem	diag1a 50028*	The constant functor of 𝑋. (Contributed by Zhi Wang, 19-Oct-2025.)
⊢ 𝐿 = (𝐶Δfunc𝐷)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ 𝐴 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ 𝐾 = ((1st ‘𝐿)‘𝑋)    &   ⊢ 𝐵 = (Base‘𝐷)    &   ⊢ 𝐽 = (Hom ‘𝐷)    &   ⊢ 1 = (Id‘𝐶)    ⇒   ⊢ (𝜑 → 𝐾 = ⟨(𝐵 × {𝑋}), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ ((𝑦𝐽𝑧) × {( 1 ‘𝑋)}))⟩)
Theorem	diag1f1lem 50029	The object part of the diagonal functor is 1-1 if 𝐵 is non-empty. Note that (𝜑 → (𝑀 = 𝑁 ↔ 𝑋 = 𝑌)) also holds because of diag1f1 50030 and f1fveq 7260. (Contributed by Zhi Wang, 19-Oct-2025.)
⊢ 𝐿 = (𝐶Δfunc𝐷)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ 𝐴 = (Base‘𝐶)    &   ⊢ 𝐵 = (Base‘𝐷)    &   ⊢ (𝜑 → 𝐵 ≠ ∅)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐴)    &   ⊢ 𝑀 = ((1st ‘𝐿)‘𝑋)    &   ⊢ 𝑁 = ((1st ‘𝐿)‘𝑌)    ⇒   ⊢ (𝜑 → (𝑀 = 𝑁 → 𝑋 = 𝑌))
Theorem	diag1f1 50030	The object part of the diagonal functor is 1-1 if 𝐵 is non-empty. (Contributed by Zhi Wang, 19-Oct-2025.)
⊢ 𝐿 = (𝐶Δfunc𝐷)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ 𝐴 = (Base‘𝐶)    &   ⊢ 𝐵 = (Base‘𝐷)    &   ⊢ (𝜑 → 𝐵 ≠ ∅)    ⇒   ⊢ (𝜑 → (1st ‘𝐿):𝐴–1-1→(𝐷 Func 𝐶))
Theorem	diag2f1lem 50031	Lemma for diag2f1 50032. The converse is trivial (fveq2 6881). (Contributed by Zhi Wang, 21-Oct-2025.)
⊢ 𝐿 = (𝐶Δfunc𝐷)    &   ⊢ 𝐴 = (Base‘𝐶)    &   ⊢ 𝐵 = (Base‘𝐷)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐴)    &   ⊢ (𝜑 → 𝐵 ≠ ∅)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))    &   ⊢ (𝜑 → 𝐺 ∈ (𝑋𝐻𝑌))    ⇒   ⊢ (𝜑 → (((𝑋(2nd ‘𝐿)𝑌)‘𝐹) = ((𝑋(2nd ‘𝐿)𝑌)‘𝐺) → 𝐹 = 𝐺))
Theorem	diag2f1 50032	If 𝐵 is non-empty, the morphism part of a diagonal functor is injective functions from hom-sets into sets of natural transformations. (Contributed by Zhi Wang, 21-Oct-2025.)
⊢ 𝐿 = (𝐶Δfunc𝐷)    &   ⊢ 𝐴 = (Base‘𝐶)    &   ⊢ 𝐵 = (Base‘𝐷)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐴)    &   ⊢ (𝜑 → 𝐵 ≠ ∅)    &   ⊢ 𝑁 = (𝐷 Nat 𝐶)    ⇒   ⊢ (𝜑 → (𝑋(2nd ‘𝐿)𝑌):(𝑋𝐻𝑌)–1-1→(((1st ‘𝐿)‘𝑋)𝑁((1st ‘𝐿)‘𝑌)))
21.50.15.18  Functor composition bifunctors
Theorem	fucofulem1 50033	Lemma for proving functor theorems. (Contributed by Zhi Wang, 25-Sep-2025.)
⊢ (𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜃 ∧ 𝜏)))    &   ⊢ ((𝜑 ∧ (𝜃 ∧ 𝜏)) → 𝜂)    &   ⊢ 𝜒    &   ⊢ ((𝜑 ∧ 𝜂) → 𝜃)    &   ⊢ ((𝜑 ∧ 𝜂) → 𝜏)    ⇒   ⊢ (𝜑 → (𝜓 ↔ 𝜂))
Theorem	fucofulem2 50034*	Lemma for proving functor theorems. Maybe consider eufnfv 7227 to prove the uniqueness of a functor. (Contributed by Zhi Wang, 25-Sep-2025.)
⊢ 𝐵 = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))    &   ⊢ 𝐻 = (Hom ‘((𝐷 FuncCat 𝐸) ×c (𝐶 FuncCat 𝐷)))    ⇒   ⊢ (𝐺 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st ‘𝑧))(𝐶 Nat 𝐸)(𝐹‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧)) ↔ (𝐺 = (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑢𝐺𝑣)) ∧ ∀𝑚 ∈ 𝐵 ∀𝑛 ∈ 𝐵 ((𝑚𝐺𝑛) = (𝑏 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛)), 𝑎 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛)) ↦ (𝑏(𝑚𝐺𝑛)𝑎)) ∧ ∀𝑝 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛))∀𝑞 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛))(𝑝(𝑚𝐺𝑛)𝑞) ∈ ((𝐹‘𝑚)(𝐶 Nat 𝐸)(𝐹‘𝑛)))))
Theorem	fuco2el 50035	Equivalence of product functor. (Contributed by Zhi Wang, 29-Sep-2025.)
⊢ (⟨⟨𝐾, 𝐿⟩, ⟨𝐹, 𝐺⟩⟩ ∈ (𝑆 × 𝑅) ↔ (𝐾𝑆𝐿 ∧ 𝐹𝑅𝐺))
Theorem	fuco2eld 50036	Equivalence of product functor. (Contributed by Zhi Wang, 29-Sep-2025.)
⊢ (𝜑 → 𝑊 = (𝑆 × 𝑅))    &   ⊢ (𝜑 → 𝑈 = ⟨⟨𝐾, 𝐿⟩, ⟨𝐹, 𝐺⟩⟩)    &   ⊢ (𝜑 → 𝐾𝑆𝐿)    &   ⊢ (𝜑 → 𝐹𝑅𝐺)    ⇒   ⊢ (𝜑 → 𝑈 ∈ 𝑊)
Theorem	fuco2eld2 50037	Equivalence of product functor. (Contributed by Zhi Wang, 29-Sep-2025.)
⊢ (𝜑 → 𝑊 = (𝑆 × 𝑅))    &   ⊢ (𝜑 → 𝑈 ∈ 𝑊)    &   ⊢ Rel 𝑆    &   ⊢ Rel 𝑅    ⇒   ⊢ (𝜑 → 𝑈 = ⟨⟨(1st ‘(1st ‘𝑈)), (2nd ‘(1st ‘𝑈))⟩, ⟨(1st ‘(2nd ‘𝑈)), (2nd ‘(2nd ‘𝑈))⟩⟩)
Theorem	fuco2eld3 50038	Equivalence of product functor. (Contributed by Zhi Wang, 29-Sep-2025.)
⊢ (𝜑 → 𝑊 = (𝑆 × 𝑅))    &   ⊢ (𝜑 → 𝑈 ∈ 𝑊)    &   ⊢ Rel 𝑆    &   ⊢ Rel 𝑅    ⇒   ⊢ (𝜑 → ((1st ‘(1st ‘𝑈))𝑆(2nd ‘(1st ‘𝑈)) ∧ (1st ‘(2nd ‘𝑈))𝑅(2nd ‘(2nd ‘𝑈))))
Syntax	cfuco 50039	Extend class notation with functor composition bifunctors.
class ∘F
Definition	df-fuco 50040*	Definition of functor composition bifunctors. Given three categories 𝐶, 𝐷, and 𝐸, (⟨𝐶, 𝐷⟩ ∘F 𝐸) is a functor from the product category of two categories of functors to a category of functors (fucofunc 50082). The object part maps two functors to their composition (fuco11 50049 and fuco11b 50060). The morphism part defines the "composition" of two natural transformations (fuco22 50062) into another natural transformation (fuco22nat 50069) such that a "cube-like" diagram commutes. The naturality property also gives an alternate definition (fuco23a 50075). Note that such "composition" is different from fucco 18021 because they "compose" along different "axes". (Contributed by Zhi Wang, 29-Sep-2025.)
⊢ ∘F = (𝑝 ∈ V, 𝑒 ∈ V ↦ ⦋(1st ‘𝑝) / 𝑐⦌⦋(2nd ‘𝑝) / 𝑑⦌⦋((𝑑 Func 𝑒) × (𝑐 Func 𝑑)) / 𝑤⦌⟨( ∘func ↾ 𝑤), (𝑢 ∈ 𝑤, 𝑣 ∈ 𝑤 ↦ ⦋(1st ‘(2nd ‘𝑢)) / 𝑓⦌⦋(1st ‘(1st ‘𝑢)) / 𝑘⦌⦋(2nd ‘(1st ‘𝑢)) / 𝑙⦌⦋(1st ‘(2nd ‘𝑣)) / 𝑚⦌⦋(1st ‘(1st ‘𝑣)) / 𝑟⦌(𝑏 ∈ ((1st ‘𝑢)(𝑑 Nat 𝑒)(1st ‘𝑣)), 𝑎 ∈ ((2nd ‘𝑢)(𝑐 Nat 𝑑)(2nd ‘𝑣)) ↦ (𝑥 ∈ (Base‘𝑐) ↦ ((𝑏‘(𝑚‘𝑥))(⟨(𝑘‘(𝑓‘𝑥)), (𝑘‘(𝑚‘𝑥))⟩(comp‘𝑒)(𝑟‘(𝑚‘𝑥)))(((𝑓‘𝑥)𝑙(𝑚‘𝑥))‘(𝑎‘𝑥))))))⟩)
Theorem	fucofvalg 50041*	Value of the function giving the functor composition bifunctor. (Contributed by Zhi Wang, 7-Oct-2025.)
⊢ (𝜑 → 𝑃 ∈ 𝑈)    &   ⊢ (𝜑 → (1st ‘𝑃) = 𝐶)    &   ⊢ (𝜑 → (2nd ‘𝑃) = 𝐷)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑃 ∘F 𝐸) = ⚬ )    &   ⊢ (𝜑 → 𝑊 = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))    ⇒   ⊢ (𝜑 → ⚬ = ⟨( ∘func ↾ 𝑊), (𝑢 ∈ 𝑊, 𝑣 ∈ 𝑊 ↦ ⦋(1st ‘(2nd ‘𝑢)) / 𝑓⦌⦋(1st ‘(1st ‘𝑢)) / 𝑘⦌⦋(2nd ‘(1st ‘𝑢)) / 𝑙⦌⦋(1st ‘(2nd ‘𝑣)) / 𝑚⦌⦋(1st ‘(1st ‘𝑣)) / 𝑟⦌(𝑏 ∈ ((1st ‘𝑢)(𝐷 Nat 𝐸)(1st ‘𝑣)), 𝑎 ∈ ((2nd ‘𝑢)(𝐶 Nat 𝐷)(2nd ‘𝑣)) ↦ (𝑥 ∈ (Base‘𝐶) ↦ ((𝑏‘(𝑚‘𝑥))(⟨(𝑘‘(𝑓‘𝑥)), (𝑘‘(𝑚‘𝑥))⟩(comp‘𝐸)(𝑟‘(𝑚‘𝑥)))(((𝑓‘𝑥)𝑙(𝑚‘𝑥))‘(𝑎‘𝑥))))))⟩)
Theorem	fucofval 50042*	Value of the function giving the functor composition bifunctor. Hypotheses fucofval.c and fucofval.d are not redundant (fucofvalne 50048). (Contributed by Zhi Wang, 29-Sep-2025.)
⊢ (𝜑 → 𝐶 ∈ 𝑇)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑉)    &   ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = ⚬ )    &   ⊢ (𝜑 → 𝑊 = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))    ⇒   ⊢ (𝜑 → ⚬ = ⟨( ∘func ↾ 𝑊), (𝑢 ∈ 𝑊, 𝑣 ∈ 𝑊 ↦ ⦋(1st ‘(2nd ‘𝑢)) / 𝑓⦌⦋(1st ‘(1st ‘𝑢)) / 𝑘⦌⦋(2nd ‘(1st ‘𝑢)) / 𝑙⦌⦋(1st ‘(2nd ‘𝑣)) / 𝑚⦌⦋(1st ‘(1st ‘𝑣)) / 𝑟⦌(𝑏 ∈ ((1st ‘𝑢)(𝐷 Nat 𝐸)(1st ‘𝑣)), 𝑎 ∈ ((2nd ‘𝑢)(𝐶 Nat 𝐷)(2nd ‘𝑣)) ↦ (𝑥 ∈ (Base‘𝐶) ↦ ((𝑏‘(𝑚‘𝑥))(⟨(𝑘‘(𝑓‘𝑥)), (𝑘‘(𝑚‘𝑥))⟩(comp‘𝐸)(𝑟‘(𝑚‘𝑥)))(((𝑓‘𝑥)𝑙(𝑚‘𝑥))‘(𝑎‘𝑥))))))⟩)
Theorem	fucoelvv 50043	A functor composition bifunctor is an ordered pair. Enables 1st2ndb 8025. (Contributed by Zhi Wang, 29-Sep-2025.)
⊢ (𝜑 → 𝐶 ∈ 𝑇)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑉)    &   ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = ⚬ )    ⇒   ⊢ (𝜑 → ⚬ ∈ (V × V))
Theorem	fuco1 50044	The object part of the functor composition bifunctor. (Contributed by Zhi Wang, 29-Sep-2025.)
⊢ (𝜑 → 𝐶 ∈ 𝑇)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑉)    &   ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = ⟨𝑂, 𝑃⟩)    &   ⊢ (𝜑 → 𝑊 = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))    ⇒   ⊢ (𝜑 → 𝑂 = ( ∘func ↾ 𝑊))
Theorem	fucof1 50045	The object part of the functor composition bifunctor maps ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) into (𝐶 Func 𝐸). (Contributed by Zhi Wang, 29-Sep-2025.)
⊢ (𝜑 → 𝐶 ∈ 𝑇)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑉)    &   ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = ⟨𝑂, 𝑃⟩)    &   ⊢ (𝜑 → 𝑊 = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))    ⇒   ⊢ (𝜑 → 𝑂:𝑊⟶(𝐶 Func 𝐸))
Theorem	fuco2 50046*	The morphism part of the functor composition bifunctor. (Contributed by Zhi Wang, 29-Sep-2025.)
⊢ (𝜑 → 𝐶 ∈ 𝑇)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑉)    &   ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = ⟨𝑂, 𝑃⟩)    &   ⊢ (𝜑 → 𝑊 = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))    ⇒   ⊢ (𝜑 → 𝑃 = (𝑢 ∈ 𝑊, 𝑣 ∈ 𝑊 ↦ ⦋(1st ‘(2nd ‘𝑢)) / 𝑓⦌⦋(1st ‘(1st ‘𝑢)) / 𝑘⦌⦋(2nd ‘(1st ‘𝑢)) / 𝑙⦌⦋(1st ‘(2nd ‘𝑣)) / 𝑚⦌⦋(1st ‘(1st ‘𝑣)) / 𝑟⦌(𝑏 ∈ ((1st ‘𝑢)(𝐷 Nat 𝐸)(1st ‘𝑣)), 𝑎 ∈ ((2nd ‘𝑢)(𝐶 Nat 𝐷)(2nd ‘𝑣)) ↦ (𝑥 ∈ (Base‘𝐶) ↦ ((𝑏‘(𝑚‘𝑥))(⟨(𝑘‘(𝑓‘𝑥)), (𝑘‘(𝑚‘𝑥))⟩(comp‘𝐸)(𝑟‘(𝑚‘𝑥)))(((𝑓‘𝑥)𝑙(𝑚‘𝑥))‘(𝑎‘𝑥)))))))
Theorem	fucofn2 50047	The morphism part of the functor composition bifunctor is a function on the Cartesian square of the base set. (Contributed by Zhi Wang, 29-Sep-2025.)
⊢ (𝜑 → 𝐶 ∈ 𝑇)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑉)    &   ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = ⟨𝑂, 𝑃⟩)    &   ⊢ (𝜑 → 𝑊 = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))    ⇒   ⊢ (𝜑 → 𝑃 Fn (𝑊 × 𝑊))
Theorem	fucofvalne 50048*	Value of the function giving the functor composition bifunctor, if 𝐶 or 𝐷 are not sets. (Contributed by Zhi Wang, 7-Oct-2025.)
⊢ (𝜑 → ¬ (𝐶 ∈ V ∧ 𝐷 ∈ V))    &   ⊢ (𝜑 → 𝐸 ∈ Cat)    &   ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = ⚬ )    &   ⊢ (𝜑 → 𝑊 = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))    ⇒   ⊢ (𝜑 → ⚬ ≠ ⟨( ∘func ↾ 𝑊), (𝑢 ∈ 𝑊, 𝑣 ∈ 𝑊 ↦ ⦋(1st ‘(2nd ‘𝑢)) / 𝑓⦌⦋(1st ‘(1st ‘𝑢)) / 𝑘⦌⦋(2nd ‘(1st ‘𝑢)) / 𝑙⦌⦋(1st ‘(2nd ‘𝑣)) / 𝑚⦌⦋(1st ‘(1st ‘𝑣)) / 𝑟⦌(𝑏 ∈ ((1st ‘𝑢)(𝐷 Nat 𝐸)(1st ‘𝑣)), 𝑎 ∈ ((2nd ‘𝑢)(𝐶 Nat 𝐷)(2nd ‘𝑣)) ↦ (𝑥 ∈ (Base‘𝐶) ↦ ((𝑏‘(𝑚‘𝑥))(⟨(𝑘‘(𝑓‘𝑥)), (𝑘‘(𝑚‘𝑥))⟩(comp‘𝐸)(𝑟‘(𝑚‘𝑥)))(((𝑓‘𝑥)𝑙(𝑚‘𝑥))‘(𝑎‘𝑥))))))⟩)
Theorem	fuco11 50049	The object part of the functor composition bifunctor maps two functors to their composition. (Contributed by Zhi Wang, 30-Sep-2025.)
⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = ⟨𝑂, 𝑃⟩)    &   ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)    &   ⊢ (𝜑 → 𝐾(𝐷 Func 𝐸)𝐿)    &   ⊢ (𝜑 → 𝑈 = ⟨⟨𝐾, 𝐿⟩, ⟨𝐹, 𝐺⟩⟩)    ⇒   ⊢ (𝜑 → (𝑂‘𝑈) = (⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩))
Theorem	fuco11cl 50050	The object part of the functor composition bifunctor maps into (𝐶 Func 𝐸). (Contributed by Zhi Wang, 30-Sep-2025.)
⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = ⟨𝑂, 𝑃⟩)    &   ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)    &   ⊢ (𝜑 → 𝐾(𝐷 Func 𝐸)𝐿)    &   ⊢ (𝜑 → 𝑈 = ⟨⟨𝐾, 𝐿⟩, ⟨𝐹, 𝐺⟩⟩)    ⇒   ⊢ (𝜑 → (𝑂‘𝑈) ∈ (𝐶 Func 𝐸))
Theorem	fuco11a 50051*	The object part of the functor composition bifunctor maps two functors to their composition, expressed explicitly. (Contributed by Zhi Wang, 30-Sep-2025.)
⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = ⟨𝑂, 𝑃⟩)    &   ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)    &   ⊢ (𝜑 → 𝐾(𝐷 Func 𝐸)𝐿)    &   ⊢ (𝜑 → 𝑈 = ⟨⟨𝐾, 𝐿⟩, ⟨𝐹, 𝐺⟩⟩)    &   ⊢ 𝐵 = (Base‘𝐶)    ⇒   ⊢ (𝜑 → (𝑂‘𝑈) = ⟨(𝐾 ∘ 𝐹), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (((𝐹‘𝑥)𝐿(𝐹‘𝑦)) ∘ (𝑥𝐺𝑦)))⟩)
Theorem	fuco112 50052*	The object part of the functor composition bifunctor maps two functors to their composition, expressed explicitly for the morphism part of the composed functor. (Contributed by Zhi Wang, 3-Oct-2025.)
⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = ⟨𝑂, 𝑃⟩)    &   ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)    &   ⊢ (𝜑 → 𝐾(𝐷 Func 𝐸)𝐿)    &   ⊢ (𝜑 → 𝑈 = ⟨⟨𝐾, 𝐿⟩, ⟨𝐹, 𝐺⟩⟩)    &   ⊢ 𝐵 = (Base‘𝐶)    ⇒   ⊢ (𝜑 → (2nd ‘(𝑂‘𝑈)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (((𝐹‘𝑥)𝐿(𝐹‘𝑦)) ∘ (𝑥𝐺𝑦))))
Theorem	fuco111 50053	The object part of the functor composition bifunctor maps two functors to their composition, expressed explicitly for the object part of the composed functor. (Contributed by Zhi Wang, 2-Oct-2025.)
⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = ⟨𝑂, 𝑃⟩)    &   ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)    &   ⊢ (𝜑 → 𝐾(𝐷 Func 𝐸)𝐿)    &   ⊢ (𝜑 → 𝑈 = ⟨⟨𝐾, 𝐿⟩, ⟨𝐹, 𝐺⟩⟩)    ⇒   ⊢ (𝜑 → (1st ‘(𝑂‘𝑈)) = (𝐾 ∘ 𝐹))
Theorem	fuco111x 50054	The object part of the functor composition bifunctor maps two functors to their composition, expressed explicitly for the object part of the composed functor. An object is mapped by two functors in succession. (Contributed by Zhi Wang, 3-Oct-2025.)
⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = ⟨𝑂, 𝑃⟩)    &   ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)    &   ⊢ (𝜑 → 𝐾(𝐷 Func 𝐸)𝐿)    &   ⊢ (𝜑 → 𝑈 = ⟨⟨𝐾, 𝐿⟩, ⟨𝐹, 𝐺⟩⟩)    &   ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))    ⇒   ⊢ (𝜑 → ((1st ‘(𝑂‘𝑈))‘𝑋) = (𝐾‘(𝐹‘𝑋)))
Theorem	fuco112x 50055	The object part of the functor composition bifunctor maps two functors to their composition, expressed explicitly for the morphism part of the composed functor. (Contributed by Zhi Wang, 3-Oct-2025.)
⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = ⟨𝑂, 𝑃⟩)    &   ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)    &   ⊢ (𝜑 → 𝐾(𝐷 Func 𝐸)𝐿)    &   ⊢ (𝜑 → 𝑈 = ⟨⟨𝐾, 𝐿⟩, ⟨𝐹, 𝐺⟩⟩)    &   ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))    &   ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐶))    ⇒   ⊢ (𝜑 → (𝑋(2nd ‘(𝑂‘𝑈))𝑌) = (((𝐹‘𝑋)𝐿(𝐹‘𝑌)) ∘ (𝑋𝐺𝑌)))
Theorem	fuco112xa 50056	The object part of the functor composition bifunctor maps two functors to their composition, expressed explicitly for the morphism part of the composed functor. A morphism is mapped by two functors in succession. (Contributed by Zhi Wang, 3-Oct-2025.)
⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = ⟨𝑂, 𝑃⟩)    &   ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)    &   ⊢ (𝜑 → 𝐾(𝐷 Func 𝐸)𝐿)    &   ⊢ (𝜑 → 𝑈 = ⟨⟨𝐾, 𝐿⟩, ⟨𝐹, 𝐺⟩⟩)    &   ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))    &   ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐶))    &   ⊢ (𝜑 → 𝐴 ∈ (𝑋(Hom ‘𝐶)𝑌))    ⇒   ⊢ (𝜑 → ((𝑋(2nd ‘(𝑂‘𝑈))𝑌)‘𝐴) = (((𝐹‘𝑋)𝐿(𝐹‘𝑌))‘((𝑋𝐺𝑌)‘𝐴)))
Theorem	fuco11id 50057	The identity morphism of the mapped object. (Contributed by Zhi Wang, 30-Sep-2025.)
⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = ⟨𝑂, 𝑃⟩)    &   ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)    &   ⊢ (𝜑 → 𝐾(𝐷 Func 𝐸)𝐿)    &   ⊢ (𝜑 → 𝑈 = ⟨⟨𝐾, 𝐿⟩, ⟨𝐹, 𝐺⟩⟩)    &   ⊢ 𝑄 = (𝐶 FuncCat 𝐸)    &   ⊢ 𝐼 = (Id‘𝑄)    &   ⊢ 1 = (Id‘𝐸)    ⇒   ⊢ (𝜑 → (𝐼‘(𝑂‘𝑈)) = ( 1 ∘ (𝐾 ∘ 𝐹)))
Theorem	fuco11idx 50058	The identity morphism of the mapped object. (Contributed by Zhi Wang, 3-Oct-2025.)
⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = ⟨𝑂, 𝑃⟩)    &   ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)    &   ⊢ (𝜑 → 𝐾(𝐷 Func 𝐸)𝐿)    &   ⊢ (𝜑 → 𝑈 = ⟨⟨𝐾, 𝐿⟩, ⟨𝐹, 𝐺⟩⟩)    &   ⊢ 𝑄 = (𝐶 FuncCat 𝐸)    &   ⊢ 𝐼 = (Id‘𝑄)    &   ⊢ 1 = (Id‘𝐸)    &   ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))    ⇒   ⊢ (𝜑 → ((𝐼‘(𝑂‘𝑈))‘𝑋) = ( 1 ‘(𝐾‘(𝐹‘𝑋))))
Theorem	fuco21 50059*	The morphism part of the functor composition bifunctor. (Contributed by Zhi Wang, 29-Sep-2025.)
⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = ⟨𝑂, 𝑃⟩)    &   ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)    &   ⊢ (𝜑 → 𝐾(𝐷 Func 𝐸)𝐿)    &   ⊢ (𝜑 → 𝑈 = ⟨⟨𝐾, 𝐿⟩, ⟨𝐹, 𝐺⟩⟩)    &   ⊢ (𝜑 → 𝑀(𝐶 Func 𝐷)𝑁)    &   ⊢ (𝜑 → 𝑅(𝐷 Func 𝐸)𝑆)    &   ⊢ (𝜑 → 𝑉 = ⟨⟨𝑅, 𝑆⟩, ⟨𝑀, 𝑁⟩⟩)    ⇒   ⊢ (𝜑 → (𝑈𝑃𝑉) = (𝑏 ∈ (⟨𝐾, 𝐿⟩(𝐷 Nat 𝐸)⟨𝑅, 𝑆⟩), 𝑎 ∈ (⟨𝐹, 𝐺⟩(𝐶 Nat 𝐷)⟨𝑀, 𝑁⟩) ↦ (𝑥 ∈ (Base‘𝐶) ↦ ((𝑏‘(𝑀‘𝑥))(⟨(𝐾‘(𝐹‘𝑥)), (𝐾‘(𝑀‘𝑥))⟩(comp‘𝐸)(𝑅‘(𝑀‘𝑥)))(((𝐹‘𝑥)𝐿(𝑀‘𝑥))‘(𝑎‘𝑥))))))
Theorem	fuco11b 50060	The object part of the functor composition bifunctor maps two functors to their composition. (Contributed by Zhi Wang, 11-Oct-2025.)
⊢ (𝜑 → (1st ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸)) = 𝑂)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    &   ⊢ (𝜑 → 𝐺 ∈ (𝐷 Func 𝐸))    ⇒   ⊢ (𝜑 → (𝐺𝑂𝐹) = (𝐺 ∘func 𝐹))
Theorem	fuco11bALT 50061	Alternate proof of fuco11b 50060. (Contributed by Zhi Wang, 11-Oct-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → (1st ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸)) = 𝑂)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    &   ⊢ (𝜑 → 𝐺 ∈ (𝐷 Func 𝐸))    ⇒   ⊢ (𝜑 → (𝐺𝑂𝐹) = (𝐺 ∘func 𝐹))
Theorem	fuco22 50062*	The morphism part of the functor composition bifunctor. See also fuco22a 50073. (Contributed by Zhi Wang, 29-Sep-2025.)
⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = ⟨𝑂, 𝑃⟩)    &   ⊢ (𝜑 → 𝑈 = ⟨⟨𝐾, 𝐿⟩, ⟨𝐹, 𝐺⟩⟩)    &   ⊢ (𝜑 → 𝑉 = ⟨⟨𝑅, 𝑆⟩, ⟨𝑀, 𝑁⟩⟩)    &   ⊢ (𝜑 → 𝐴 ∈ (⟨𝐹, 𝐺⟩(𝐶 Nat 𝐷)⟨𝑀, 𝑁⟩))    &   ⊢ (𝜑 → 𝐵 ∈ (⟨𝐾, 𝐿⟩(𝐷 Nat 𝐸)⟨𝑅, 𝑆⟩))    ⇒   ⊢ (𝜑 → (𝐵(𝑈𝑃𝑉)𝐴) = (𝑥 ∈ (Base‘𝐶) ↦ ((𝐵‘(𝑀‘𝑥))(⟨(𝐾‘(𝐹‘𝑥)), (𝐾‘(𝑀‘𝑥))⟩(comp‘𝐸)(𝑅‘(𝑀‘𝑥)))(((𝐹‘𝑥)𝐿(𝑀‘𝑥))‘(𝐴‘𝑥)))))
Theorem	fucofn22 50063	The morphism part of the functor composition bifunctor maps two natural transformations to a function on a base set. (Contributed by Zhi Wang, 30-Sep-2025.)
⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = ⟨𝑂, 𝑃⟩)    &   ⊢ (𝜑 → 𝑈 = ⟨⟨𝐾, 𝐿⟩, ⟨𝐹, 𝐺⟩⟩)    &   ⊢ (𝜑 → 𝑉 = ⟨⟨𝑅, 𝑆⟩, ⟨𝑀, 𝑁⟩⟩)    &   ⊢ (𝜑 → 𝐴 ∈ (⟨𝐹, 𝐺⟩(𝐶 Nat 𝐷)⟨𝑀, 𝑁⟩))    &   ⊢ (𝜑 → 𝐵 ∈ (⟨𝐾, 𝐿⟩(𝐷 Nat 𝐸)⟨𝑅, 𝑆⟩))    ⇒   ⊢ (𝜑 → (𝐵(𝑈𝑃𝑉)𝐴) Fn (Base‘𝐶))
Theorem	fuco23 50064	The morphism part of the functor composition bifunctor. See also fuco23a 50075. (Contributed by Zhi Wang, 29-Sep-2025.)
⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = ⟨𝑂, 𝑃⟩)    &   ⊢ (𝜑 → 𝑈 = ⟨⟨𝐾, 𝐿⟩, ⟨𝐹, 𝐺⟩⟩)    &   ⊢ (𝜑 → 𝑉 = ⟨⟨𝑅, 𝑆⟩, ⟨𝑀, 𝑁⟩⟩)    &   ⊢ (𝜑 → 𝐴 ∈ (⟨𝐹, 𝐺⟩(𝐶 Nat 𝐷)⟨𝑀, 𝑁⟩))    &   ⊢ (𝜑 → 𝐵 ∈ (⟨𝐾, 𝐿⟩(𝐷 Nat 𝐸)⟨𝑅, 𝑆⟩))    &   ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))    &   ⊢ (𝜑 → ∗ = (⟨(𝐾‘(𝐹‘𝑋)), (𝐾‘(𝑀‘𝑋))⟩(comp‘𝐸)(𝑅‘(𝑀‘𝑋))))    ⇒   ⊢ (𝜑 → ((𝐵(𝑈𝑃𝑉)𝐴)‘𝑋) = ((𝐵‘(𝑀‘𝑋)) ∗ (((𝐹‘𝑋)𝐿(𝑀‘𝑋))‘(𝐴‘𝑋))))
Theorem	fuco22natlem1 50065	Lemma for fuco22nat 50069. The commutative square of natural transformation 𝐴 in category 𝐷, mapped to category 𝐸 by the morphism part 𝐿 of the functor. (Contributed by Zhi Wang, 30-Sep-2025.)
⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))    &   ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐶))    &   ⊢ (𝜑 → 𝐴 ∈ (⟨𝐹, 𝐺⟩(𝐶 Nat 𝐷)⟨𝑀, 𝑁⟩))    &   ⊢ (𝜑 → 𝐻 ∈ (𝑋(Hom ‘𝐶)𝑌))    &   ⊢ (𝜑 → 𝐾(𝐷 Func 𝐸)𝐿)    ⇒   ⊢ (𝜑 → ((((𝐹‘𝑌)𝐿(𝑀‘𝑌))‘(𝐴‘𝑌))(⟨(𝐾‘(𝐹‘𝑋)), (𝐾‘(𝐹‘𝑌))⟩(comp‘𝐸)(𝐾‘(𝑀‘𝑌)))(((𝐹‘𝑋)𝐿(𝐹‘𝑌))‘((𝑋𝐺𝑌)‘𝐻))) = ((((𝑀‘𝑋)𝐿(𝑀‘𝑌))‘((𝑋𝑁𝑌)‘𝐻))(⟨(𝐾‘(𝐹‘𝑋)), (𝐾‘(𝑀‘𝑋))⟩(comp‘𝐸)(𝐾‘(𝑀‘𝑌)))(((𝐹‘𝑋)𝐿(𝑀‘𝑋))‘(𝐴‘𝑋))))
Theorem	fuco22natlem2 50066	Lemma for fuco22nat 50069. The commutative square of natural transformation 𝐵 in category 𝐸, combined with the commutative square of fuco22natlem1 50065. (Contributed by Zhi Wang, 30-Sep-2025.)
⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))    &   ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐶))    &   ⊢ (𝜑 → 𝐴 ∈ (⟨𝐹, 𝐺⟩(𝐶 Nat 𝐷)⟨𝑀, 𝑁⟩))    &   ⊢ (𝜑 → 𝐻 ∈ (𝑋(Hom ‘𝐶)𝑌))    &   ⊢ (𝜑 → 𝐵 ∈ (⟨𝐾, 𝐿⟩(𝐷 Nat 𝐸)⟨𝑅, 𝑆⟩))    ⇒   ⊢ (𝜑 → (((𝐵‘(𝑀‘𝑌))(⟨(𝐾‘(𝐹‘𝑌)), (𝐾‘(𝑀‘𝑌))⟩(comp‘𝐸)(𝑅‘(𝑀‘𝑌)))(((𝐹‘𝑌)𝐿(𝑀‘𝑌))‘(𝐴‘𝑌)))(⟨(𝐾‘(𝐹‘𝑋)), (𝐾‘(𝐹‘𝑌))⟩(comp‘𝐸)(𝑅‘(𝑀‘𝑌)))(((𝐹‘𝑋)𝐿(𝐹‘𝑌))‘((𝑋𝐺𝑌)‘𝐻))) = ((((𝑀‘𝑋)𝑆(𝑀‘𝑌))‘((𝑋𝑁𝑌)‘𝐻))(⟨(𝐾‘(𝐹‘𝑋)), (𝑅‘(𝑀‘𝑋))⟩(comp‘𝐸)(𝑅‘(𝑀‘𝑌)))((𝐵‘(𝑀‘𝑋))(⟨(𝐾‘(𝐹‘𝑋)), (𝐾‘(𝑀‘𝑋))⟩(comp‘𝐸)(𝑅‘(𝑀‘𝑋)))(((𝐹‘𝑋)𝐿(𝑀‘𝑋))‘(𝐴‘𝑋)))))
Theorem	fuco22natlem3 50067	Combine fuco22natlem2 50066 with fuco23 50064. (Contributed by Zhi Wang, 30-Sep-2025.)
⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))    &   ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐶))    &   ⊢ (𝜑 → 𝐴 ∈ (⟨𝐹, 𝐺⟩(𝐶 Nat 𝐷)⟨𝑀, 𝑁⟩))    &   ⊢ (𝜑 → 𝐻 ∈ (𝑋(Hom ‘𝐶)𝑌))    &   ⊢ (𝜑 → 𝐵 ∈ (⟨𝐾, 𝐿⟩(𝐷 Nat 𝐸)⟨𝑅, 𝑆⟩))    &   ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = ⟨𝑂, 𝑃⟩)    &   ⊢ (𝜑 → 𝑈 = ⟨⟨𝐾, 𝐿⟩, ⟨𝐹, 𝐺⟩⟩)    &   ⊢ (𝜑 → 𝑉 = ⟨⟨𝑅, 𝑆⟩, ⟨𝑀, 𝑁⟩⟩)    ⇒   ⊢ (𝜑 → (((𝐵(𝑈𝑃𝑉)𝐴)‘𝑌)(⟨((𝐾 ∘ 𝐹)‘𝑋), ((𝐾 ∘ 𝐹)‘𝑌)⟩(comp‘𝐸)((𝑅 ∘ 𝑀)‘𝑌))((((𝐹‘𝑋)𝐿(𝐹‘𝑌)) ∘ (𝑋𝐺𝑌))‘𝐻)) = (((((𝑀‘𝑋)𝑆(𝑀‘𝑌)) ∘ (𝑋𝑁𝑌))‘𝐻)(⟨((𝐾 ∘ 𝐹)‘𝑋), ((𝑅 ∘ 𝑀)‘𝑋)⟩(comp‘𝐸)((𝑅 ∘ 𝑀)‘𝑌))((𝐵(𝑈𝑃𝑉)𝐴)‘𝑋)))
Theorem	fuco22natlem 50068	The composed natural transformation is a natural transformation. Use fuco22nat 50069 instead. (New usage is discouraged.) (Contributed by Zhi Wang, 30-Sep-2025.)
⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = ⟨𝑂, 𝑃⟩)    &   ⊢ (𝜑 → 𝐴 ∈ (⟨𝐹, 𝐺⟩(𝐶 Nat 𝐷)⟨𝑀, 𝑁⟩))    &   ⊢ (𝜑 → 𝐵 ∈ (⟨𝐾, 𝐿⟩(𝐷 Nat 𝐸)⟨𝑅, 𝑆⟩))    &   ⊢ (𝜑 → 𝑈 = ⟨⟨𝐾, 𝐿⟩, ⟨𝐹, 𝐺⟩⟩)    &   ⊢ (𝜑 → 𝑉 = ⟨⟨𝑅, 𝑆⟩, ⟨𝑀, 𝑁⟩⟩)    ⇒   ⊢ (𝜑 → (𝐵(𝑈𝑃𝑉)𝐴) ∈ ((𝑂‘𝑈)(𝐶 Nat 𝐸)(𝑂‘𝑉)))
Theorem	fuco22nat 50069	The composed natural transformation is a natural transformation. (Contributed by Zhi Wang, 2-Oct-2025.)
⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = ⟨𝑂, 𝑃⟩)    &   ⊢ (𝜑 → 𝐴 ∈ (𝐹(𝐶 Nat 𝐷)𝑀))    &   ⊢ (𝜑 → 𝐵 ∈ (𝐾(𝐷 Nat 𝐸)𝑅))    &   ⊢ (𝜑 → 𝑈 = ⟨𝐾, 𝐹⟩)    &   ⊢ (𝜑 → 𝑉 = ⟨𝑅, 𝑀⟩)    ⇒   ⊢ (𝜑 → (𝐵(𝑈𝑃𝑉)𝐴) ∈ ((𝑂‘𝑈)(𝐶 Nat 𝐸)(𝑂‘𝑉)))
Theorem	fucof21 50070	The morphism part of the functor composition bifunctor maps a hom-set of the product category into a set of natural transformations. (Contributed by Zhi Wang, 30-Sep-2025.)
⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = ⟨𝑂, 𝑃⟩)    &   ⊢ 𝑇 = ((𝐷 FuncCat 𝐸) ×c (𝐶 FuncCat 𝐷))    &   ⊢ 𝐽 = (Hom ‘𝑇)    &   ⊢ (𝜑 → 𝑊 = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))    &   ⊢ (𝜑 → 𝑈 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑉 ∈ 𝑊)    ⇒   ⊢ (𝜑 → (𝑈𝑃𝑉):(𝑈𝐽𝑉)⟶((𝑂‘𝑈)(𝐶 Nat 𝐸)(𝑂‘𝑉)))
Theorem	fucoid 50071	Each identity morphism in the source category is mapped to the corresponding identity morphism in the target category. See also fucoid2 50072. (Contributed by Zhi Wang, 30-Sep-2025.)
⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = ⟨𝑂, 𝑃⟩)    &   ⊢ 𝑇 = ((𝐷 FuncCat 𝐸) ×c (𝐶 FuncCat 𝐷))    &   ⊢ 1 = (Id‘𝑇)    &   ⊢ 𝑄 = (𝐶 FuncCat 𝐸)    &   ⊢ 𝐼 = (Id‘𝑄)    &   ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)    &   ⊢ (𝜑 → 𝐾(𝐷 Func 𝐸)𝐿)    &   ⊢ (𝜑 → 𝑈 = ⟨⟨𝐾, 𝐿⟩, ⟨𝐹, 𝐺⟩⟩)    ⇒   ⊢ (𝜑 → ((𝑈𝑃𝑈)‘( 1 ‘𝑈)) = (𝐼‘(𝑂‘𝑈)))
Theorem	fucoid2 50072	Each identity morphism in the source category is mapped to the corresponding identity morphism in the target category. See also fucoid 50071. (Contributed by Zhi Wang, 30-Sep-2025.)
⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = ⟨𝑂, 𝑃⟩)    &   ⊢ 𝑇 = ((𝐷 FuncCat 𝐸) ×c (𝐶 FuncCat 𝐷))    &   ⊢ 1 = (Id‘𝑇)    &   ⊢ 𝑄 = (𝐶 FuncCat 𝐸)    &   ⊢ 𝐼 = (Id‘𝑄)    &   ⊢ (𝜑 → 𝑊 = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))    &   ⊢ (𝜑 → 𝑈 ∈ 𝑊)    ⇒   ⊢ (𝜑 → ((𝑈𝑃𝑈)‘( 1 ‘𝑈)) = (𝐼‘(𝑂‘𝑈)))
Theorem	fuco22a 50073*	The morphism part of the functor composition bifunctor. See also fuco22 50062. (Contributed by Zhi Wang, 1-Oct-2025.)
⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = ⟨𝑂, 𝑃⟩)    &   ⊢ (𝜑 → 𝑈 = ⟨𝐾, 𝐹⟩)    &   ⊢ (𝜑 → 𝑉 = ⟨𝑅, 𝑀⟩)    &   ⊢ (𝜑 → 𝐴 ∈ (𝐹(𝐶 Nat 𝐷)𝑀))    &   ⊢ (𝜑 → 𝐵 ∈ (𝐾(𝐷 Nat 𝐸)𝑅))    ⇒   ⊢ (𝜑 → (𝐵(𝑈𝑃𝑉)𝐴) = (𝑥 ∈ (Base‘𝐶) ↦ ((𝐵‘((1st ‘𝑀)‘𝑥))(⟨((1st ‘𝐾)‘((1st ‘𝐹)‘𝑥)), ((1st ‘𝐾)‘((1st ‘𝑀)‘𝑥))⟩(comp‘𝐸)((1st ‘𝑅)‘((1st ‘𝑀)‘𝑥)))((((1st ‘𝐹)‘𝑥)(2nd ‘𝐾)((1st ‘𝑀)‘𝑥))‘(𝐴‘𝑥)))))
Theorem	fuco23alem 50074	The naturality property (nati 18014) in category 𝐸. (Contributed by Zhi Wang, 3-Oct-2025.)
⊢ (𝜑 → 𝐴 ∈ (⟨𝐹, 𝐺⟩(𝐶 Nat 𝐷)⟨𝑀, 𝑁⟩))    &   ⊢ (𝜑 → 𝐵 ∈ (⟨𝐾, 𝐿⟩(𝐷 Nat 𝐸)⟨𝑅, 𝑆⟩))    &   ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))    &   ⊢ · = (comp‘𝐸)    ⇒   ⊢ (𝜑 → ((𝐵‘(𝑀‘𝑋))(⟨(𝐾‘(𝐹‘𝑋)), (𝐾‘(𝑀‘𝑋))⟩ · (𝑅‘(𝑀‘𝑋)))(((𝐹‘𝑋)𝐿(𝑀‘𝑋))‘(𝐴‘𝑋))) = ((((𝐹‘𝑋)𝑆(𝑀‘𝑋))‘(𝐴‘𝑋))(⟨(𝐾‘(𝐹‘𝑋)), (𝑅‘(𝐹‘𝑋))⟩ · (𝑅‘(𝑀‘𝑋)))(𝐵‘(𝐹‘𝑋))))
Theorem	fuco23a 50075	The morphism part of the functor composition bifunctor. An alternate definition of ∘F. See also fuco23 50064. (Contributed by Zhi Wang, 3-Oct-2025.)
⊢ (𝜑 → 𝐴 ∈ (⟨𝐹, 𝐺⟩(𝐶 Nat 𝐷)⟨𝑀, 𝑁⟩))    &   ⊢ (𝜑 → 𝐵 ∈ (⟨𝐾, 𝐿⟩(𝐷 Nat 𝐸)⟨𝑅, 𝑆⟩))    &   ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))    &   ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = ⟨𝑂, 𝑃⟩)    &   ⊢ (𝜑 → 𝑈 = ⟨⟨𝐾, 𝐿⟩, ⟨𝐹, 𝐺⟩⟩)    &   ⊢ (𝜑 → 𝑉 = ⟨⟨𝑅, 𝑆⟩, ⟨𝑀, 𝑁⟩⟩)    &   ⊢ (𝜑 → ∗ = (⟨(𝐾‘(𝐹‘𝑋)), (𝑅‘(𝐹‘𝑋))⟩(comp‘𝐸)(𝑅‘(𝑀‘𝑋))))    ⇒   ⊢ (𝜑 → ((𝐵(𝑈𝑃𝑉)𝐴)‘𝑋) = ((((𝐹‘𝑋)𝑆(𝑀‘𝑋))‘(𝐴‘𝑋)) ∗ (𝐵‘(𝐹‘𝑋))))
Theorem	fucocolem1 50076	Lemma for fucoco 50080. Associativity for morphisms in category 𝐸. To simply put, ((𝑎 · 𝑏) · (𝑐 · 𝑑)) = (𝑎 · ((𝑏 · 𝑐) · 𝑑)) for morphism compositions. (Contributed by Zhi Wang, 2-Oct-2025.)
⊢ (𝜑 → 𝑅 ∈ (𝐹(𝐷 Nat 𝐸)𝐾))    &   ⊢ (𝜑 → 𝑆 ∈ (𝐺(𝐶 Nat 𝐷)𝐿))    &   ⊢ (𝜑 → 𝑈 ∈ (𝐾(𝐷 Nat 𝐸)𝑀))    &   ⊢ (𝜑 → 𝑉 ∈ (𝐿(𝐶 Nat 𝐷)𝑁))    &   ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))    &   ⊢ (𝜑 → 𝑃 ∈ (𝐷 Func 𝐸))    &   ⊢ (𝜑 → 𝑄 ∈ (𝐶 Func 𝐷))    &   ⊢ (𝜑 → 𝐴 ∈ (((1st ‘𝑃)‘((1st ‘𝑄)‘𝑋))(Hom ‘𝐸)((1st ‘𝐾)‘((1st ‘𝑁)‘𝑋))))    &   ⊢ (𝜑 → 𝐵 ∈ (((1st ‘𝐹)‘((1st ‘𝐿)‘𝑋))(Hom ‘𝐸)((1st ‘𝑃)‘((1st ‘𝑄)‘𝑋))))    ⇒   ⊢ (𝜑 → (((𝑈‘((1st ‘𝑁)‘𝑋))(⟨((1st ‘𝑃)‘((1st ‘𝑄)‘𝑋)), ((1st ‘𝐾)‘((1st ‘𝑁)‘𝑋))⟩(comp‘𝐸)((1st ‘𝑀)‘((1st ‘𝑁)‘𝑋)))𝐴)(⟨((1st ‘𝐹)‘((1st ‘𝐺)‘𝑋)), ((1st ‘𝑃)‘((1st ‘𝑄)‘𝑋))⟩(comp‘𝐸)((1st ‘𝑀)‘((1st ‘𝑁)‘𝑋)))(𝐵(⟨((1st ‘𝐹)‘((1st ‘𝐺)‘𝑋)), ((1st ‘𝐹)‘((1st ‘𝐿)‘𝑋))⟩(comp‘𝐸)((1st ‘𝑃)‘((1st ‘𝑄)‘𝑋)))((((1st ‘𝐺)‘𝑋)(2nd ‘𝐹)((1st ‘𝐿)‘𝑋))‘(𝑆‘𝑋)))) = ((𝑈‘((1st ‘𝑁)‘𝑋))(⟨((1st ‘𝐹)‘((1st ‘𝐺)‘𝑋)), ((1st ‘𝐾)‘((1st ‘𝑁)‘𝑋))⟩(comp‘𝐸)((1st ‘𝑀)‘((1st ‘𝑁)‘𝑋)))((𝐴(⟨((1st ‘𝐹)‘((1st ‘𝐿)‘𝑋)), ((1st ‘𝑃)‘((1st ‘𝑄)‘𝑋))⟩(comp‘𝐸)((1st ‘𝐾)‘((1st ‘𝑁)‘𝑋)))𝐵)(⟨((1st ‘𝐹)‘((1st ‘𝐺)‘𝑋)), ((1st ‘𝐹)‘((1st ‘𝐿)‘𝑋))⟩(comp‘𝐸)((1st ‘𝐾)‘((1st ‘𝑁)‘𝑋)))((((1st ‘𝐺)‘𝑋)(2nd ‘𝐹)((1st ‘𝐿)‘𝑋))‘(𝑆‘𝑋)))))
Theorem	fucocolem2 50077*	Lemma for fucoco 50080. The composed natural transformations are mapped to composition of 4 natural transformations. (Contributed by Zhi Wang, 2-Oct-2025.)
⊢ (𝜑 → 𝑅 ∈ (𝐹(𝐷 Nat 𝐸)𝐾))    &   ⊢ (𝜑 → 𝑆 ∈ (𝐺(𝐶 Nat 𝐷)𝐿))    &   ⊢ (𝜑 → 𝑈 ∈ (𝐾(𝐷 Nat 𝐸)𝑀))    &   ⊢ (𝜑 → 𝑉 ∈ (𝐿(𝐶 Nat 𝐷)𝑁))    &   ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = ⟨𝑂, 𝑃⟩)    &   ⊢ (𝜑 → 𝑋 = ⟨𝐹, 𝐺⟩)    &   ⊢ (𝜑 → 𝑌 = ⟨𝐾, 𝐿⟩)    &   ⊢ (𝜑 → 𝑍 = ⟨𝑀, 𝑁⟩)    &   ⊢ (𝜑 → 𝐴 = ⟨𝑅, 𝑆⟩)    &   ⊢ (𝜑 → 𝐵 = ⟨𝑈, 𝑉⟩)    &   ⊢ 𝑇 = ((𝐷 FuncCat 𝐸) ×c (𝐶 FuncCat 𝐷))    &   ⊢ · = (comp‘𝑇)    &   ⊢ ∗ = (comp‘𝐷)    ⇒   ⊢ (𝜑 → ((𝑋𝑃𝑍)‘(𝐵(⟨𝑋, 𝑌⟩ · 𝑍)𝐴)) = (𝑥 ∈ (Base‘𝐶) ↦ (((𝑈‘((1st ‘𝑁)‘𝑥))(⟨((1st ‘𝐹)‘((1st ‘𝑁)‘𝑥)), ((1st ‘𝐾)‘((1st ‘𝑁)‘𝑥))⟩(comp‘𝐸)((1st ‘𝑀)‘((1st ‘𝑁)‘𝑥)))(𝑅‘((1st ‘𝑁)‘𝑥)))(⟨((1st ‘𝐹)‘((1st ‘𝐺)‘𝑥)), ((1st ‘𝐹)‘((1st ‘𝑁)‘𝑥))⟩(comp‘𝐸)((1st ‘𝑀)‘((1st ‘𝑁)‘𝑥)))((((1st ‘𝐺)‘𝑥)(2nd ‘𝐹)((1st ‘𝑁)‘𝑥))‘((𝑉‘𝑥)(⟨((1st ‘𝐺)‘𝑥), ((1st ‘𝐿)‘𝑥)⟩ ∗ ((1st ‘𝑁)‘𝑥))(𝑆‘𝑥))))))
Theorem	fucocolem3 50078*	Lemma for fucoco 50080. The composed natural transformations are mapped to composition of 4 natural transformations. (Contributed by Zhi Wang, 3-Oct-2025.)
⊢ (𝜑 → 𝑅 ∈ (𝐹(𝐷 Nat 𝐸)𝐾))    &   ⊢ (𝜑 → 𝑆 ∈ (𝐺(𝐶 Nat 𝐷)𝐿))    &   ⊢ (𝜑 → 𝑈 ∈ (𝐾(𝐷 Nat 𝐸)𝑀))    &   ⊢ (𝜑 → 𝑉 ∈ (𝐿(𝐶 Nat 𝐷)𝑁))    &   ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = ⟨𝑂, 𝑃⟩)    &   ⊢ (𝜑 → 𝑋 = ⟨𝐹, 𝐺⟩)    &   ⊢ (𝜑 → 𝑌 = ⟨𝐾, 𝐿⟩)    &   ⊢ (𝜑 → 𝑍 = ⟨𝑀, 𝑁⟩)    &   ⊢ (𝜑 → 𝐴 = ⟨𝑅, 𝑆⟩)    &   ⊢ (𝜑 → 𝐵 = ⟨𝑈, 𝑉⟩)    &   ⊢ 𝑇 = ((𝐷 FuncCat 𝐸) ×c (𝐶 FuncCat 𝐷))    &   ⊢ · = (comp‘𝑇)    &   ⊢ ∗ = (comp‘𝐷)    ⇒   ⊢ (𝜑 → ((𝑋𝑃𝑍)‘(𝐵(⟨𝑋, 𝑌⟩ · 𝑍)𝐴)) = (𝑥 ∈ (Base‘𝐶) ↦ ((𝑈‘((1st ‘𝑁)‘𝑥))(⟨((1st ‘𝐹)‘((1st ‘𝐺)‘𝑥)), ((1st ‘𝐾)‘((1st ‘𝑁)‘𝑥))⟩(comp‘𝐸)((1st ‘𝑀)‘((1st ‘𝑁)‘𝑥)))(((𝑅‘((1st ‘𝑁)‘𝑥))(⟨((1st ‘𝐹)‘((1st ‘𝐿)‘𝑥)), ((1st ‘𝐹)‘((1st ‘𝑁)‘𝑥))⟩(comp‘𝐸)((1st ‘𝐾)‘((1st ‘𝑁)‘𝑥)))((((1st ‘𝐿)‘𝑥)(2nd ‘𝐹)((1st ‘𝑁)‘𝑥))‘(𝑉‘𝑥)))(⟨((1st ‘𝐹)‘((1st ‘𝐺)‘𝑥)), ((1st ‘𝐹)‘((1st ‘𝐿)‘𝑥))⟩(comp‘𝐸)((1st ‘𝐾)‘((1st ‘𝑁)‘𝑥)))((((1st ‘𝐺)‘𝑥)(2nd ‘𝐹)((1st ‘𝐿)‘𝑥))‘(𝑆‘𝑥))))))
Theorem	fucocolem4 50079*	Lemma for fucoco 50080. The composed natural transformations are mapped to composition of 4 natural transformations. (Contributed by Zhi Wang, 2-Oct-2025.)
⊢ (𝜑 → 𝑅 ∈ (𝐹(𝐷 Nat 𝐸)𝐾))    &   ⊢ (𝜑 → 𝑆 ∈ (𝐺(𝐶 Nat 𝐷)𝐿))    &   ⊢ (𝜑 → 𝑈 ∈ (𝐾(𝐷 Nat 𝐸)𝑀))    &   ⊢ (𝜑 → 𝑉 ∈ (𝐿(𝐶 Nat 𝐷)𝑁))    &   ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = ⟨𝑂, 𝑃⟩)    &   ⊢ (𝜑 → 𝑋 = ⟨𝐹, 𝐺⟩)    &   ⊢ (𝜑 → 𝑌 = ⟨𝐾, 𝐿⟩)    &   ⊢ (𝜑 → 𝑍 = ⟨𝑀, 𝑁⟩)    &   ⊢ (𝜑 → 𝐴 = ⟨𝑅, 𝑆⟩)    &   ⊢ (𝜑 → 𝐵 = ⟨𝑈, 𝑉⟩)    &   ⊢ 𝑄 = (𝐶 FuncCat 𝐸)    &   ⊢ ∙ = (comp‘𝑄)    ⇒   ⊢ (𝜑 → (((𝑌𝑃𝑍)‘𝐵)(⟨(𝑂‘𝑋), (𝑂‘𝑌)⟩ ∙ (𝑂‘𝑍))((𝑋𝑃𝑌)‘𝐴)) = (𝑥 ∈ (Base‘𝐶) ↦ (((𝑈‘((1st ‘𝑁)‘𝑥))(⟨((1st ‘𝐾)‘((1st ‘𝐿)‘𝑥)), ((1st ‘𝐾)‘((1st ‘𝑁)‘𝑥))⟩(comp‘𝐸)((1st ‘𝑀)‘((1st ‘𝑁)‘𝑥)))((((1st ‘𝐿)‘𝑥)(2nd ‘𝐾)((1st ‘𝑁)‘𝑥))‘(𝑉‘𝑥)))(⟨((1st ‘𝐹)‘((1st ‘𝐺)‘𝑥)), ((1st ‘𝐾)‘((1st ‘𝐿)‘𝑥))⟩(comp‘𝐸)((1st ‘𝑀)‘((1st ‘𝑁)‘𝑥)))((𝑅‘((1st ‘𝐿)‘𝑥))(⟨((1st ‘𝐹)‘((1st ‘𝐺)‘𝑥)), ((1st ‘𝐹)‘((1st ‘𝐿)‘𝑥))⟩(comp‘𝐸)((1st ‘𝐾)‘((1st ‘𝐿)‘𝑥)))((((1st ‘𝐺)‘𝑥)(2nd ‘𝐹)((1st ‘𝐿)‘𝑥))‘(𝑆‘𝑥))))))
Theorem	fucoco 50080	Composition in the source category is mapped to composition in the target. See also fucoco2 50081. (Contributed by Zhi Wang, 3-Oct-2025.)
⊢ (𝜑 → 𝑅 ∈ (𝐹(𝐷 Nat 𝐸)𝐾))    &   ⊢ (𝜑 → 𝑆 ∈ (𝐺(𝐶 Nat 𝐷)𝐿))    &   ⊢ (𝜑 → 𝑈 ∈ (𝐾(𝐷 Nat 𝐸)𝑀))    &   ⊢ (𝜑 → 𝑉 ∈ (𝐿(𝐶 Nat 𝐷)𝑁))    &   ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = ⟨𝑂, 𝑃⟩)    &   ⊢ (𝜑 → 𝑋 = ⟨𝐹, 𝐺⟩)    &   ⊢ (𝜑 → 𝑌 = ⟨𝐾, 𝐿⟩)    &   ⊢ (𝜑 → 𝑍 = ⟨𝑀, 𝑁⟩)    &   ⊢ (𝜑 → 𝐴 = ⟨𝑅, 𝑆⟩)    &   ⊢ (𝜑 → 𝐵 = ⟨𝑈, 𝑉⟩)    &   ⊢ 𝑄 = (𝐶 FuncCat 𝐸)    &   ⊢ ∙ = (comp‘𝑄)    &   ⊢ 𝑇 = ((𝐷 FuncCat 𝐸) ×c (𝐶 FuncCat 𝐷))    &   ⊢ · = (comp‘𝑇)    ⇒   ⊢ (𝜑 → ((𝑋𝑃𝑍)‘(𝐵(⟨𝑋, 𝑌⟩ · 𝑍)𝐴)) = (((𝑌𝑃𝑍)‘𝐵)(⟨(𝑂‘𝑋), (𝑂‘𝑌)⟩ ∙ (𝑂‘𝑍))((𝑋𝑃𝑌)‘𝐴)))
Theorem	fucoco2 50081	Composition in the source category is mapped to composition in the target. See also fucoco 50080. (Contributed by Zhi Wang, 3-Oct-2025.)
⊢ 𝑇 = ((𝐷 FuncCat 𝐸) ×c (𝐶 FuncCat 𝐷))    &   ⊢ 𝑄 = (𝐶 FuncCat 𝐸)    &   ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = ⟨𝑂, 𝑃⟩)    &   ⊢ · = (comp‘𝑇)    &   ⊢ ∙ = (comp‘𝑄)    &   ⊢ (𝜑 → 𝑊 = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑊)    &   ⊢ 𝐽 = (Hom ‘𝑇)    &   ⊢ (𝜑 → 𝐴 ∈ (𝑋𝐽𝑌))    &   ⊢ (𝜑 → 𝐵 ∈ (𝑌𝐽𝑍))    ⇒   ⊢ (𝜑 → ((𝑋𝑃𝑍)‘(𝐵(⟨𝑋, 𝑌⟩ · 𝑍)𝐴)) = (((𝑌𝑃𝑍)‘𝐵)(⟨(𝑂‘𝑋), (𝑂‘𝑌)⟩ ∙ (𝑂‘𝑍))((𝑋𝑃𝑌)‘𝐴)))
Theorem	fucofunc 50082	The functor composition bifunctor is a functor. See also fucofunca 50083. However, it is unlikely the unique functor compatible with the functor composition. As a counterexample, let 𝐶 and 𝐷 be terminal categories (categories of one object and one morphism, df-termc 50196), for example, (SetCat‘1o) (the trivial category, setc1oterm 50214), and 𝐸 be a category with two objects equipped with only two non-identity morphisms 𝑓 and 𝑔, pointing in the same direction. It is possible to map the ordered pair of natural transformations ⟨𝑎, 𝑖⟩, where 𝑎 sends to 𝑓 and 𝑖 is the identity natural transformation, to the other natural transformation 𝑏 sending to 𝑔, i.e., define the morphism part 𝑃 such that (𝑎(𝑈𝑃𝑉)𝑖) = 𝑏 such that (𝑏‘𝑋) = 𝑔 given hypotheses of fuco23 50064. Such construction should be provable as a functor. Given any 𝑃, it is a morphism part of a functor compatible with the object part, i.e., the functor composition, i.e., the restriction of ∘func, iff both of the following hold. 1. It has the same form as df-fuco 50040 up to fuco23 50064, but ((𝐵(𝑈𝑃𝑉)𝐴)‘𝑋) might be mapped to a different morphism in category 𝐸. See fucofulem2 50034 for some insights. 2. fuco22nat 50069, fucoid 50071, and fucoco 50080 are satisfied. (Contributed by Zhi Wang, 3-Oct-2025.)
⊢ 𝑇 = ((𝐷 FuncCat 𝐸) ×c (𝐶 FuncCat 𝐷))    &   ⊢ 𝑄 = (𝐶 FuncCat 𝐸)    &   ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = ⟨𝑂, 𝑃⟩)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ (𝜑 → 𝐸 ∈ Cat)    ⇒   ⊢ (𝜑 → 𝑂(𝑇 Func 𝑄)𝑃)
Theorem	fucofunca 50083	The functor composition bifunctor is a functor. See also fucofunc 50082. (Contributed by Zhi Wang, 10-Oct-2025.)
⊢ 𝑇 = ((𝐷 FuncCat 𝐸) ×c (𝐶 FuncCat 𝐷))    &   ⊢ 𝑄 = (𝐶 FuncCat 𝐸)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ (𝜑 → 𝐸 ∈ Cat)    ⇒   ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) ∈ (𝑇 Func 𝑄))
Theorem	fucolid 50084*	Post-compose a natural transformation with an identity natural transformation. (Contributed by Zhi Wang, 11-Oct-2025.)
⊢ (𝜑 → (2nd ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸)) = 𝑃)    &   ⊢ 𝐼 = (Id‘𝑄)    &   ⊢ 𝑄 = (𝐷 FuncCat 𝐸)    &   ⊢ (𝜑 → 𝐴 ∈ (𝐺(𝐶 Nat 𝐷)𝐻))    &   ⊢ (𝜑 → 𝐹 ∈ (𝐷 Func 𝐸))    ⇒   ⊢ (𝜑 → ((𝐼‘𝐹)(⟨𝐹, 𝐺⟩𝑃⟨𝐹, 𝐻⟩)𝐴) = (𝑥 ∈ (Base‘𝐶) ↦ ((((1st ‘𝐺)‘𝑥)(2nd ‘𝐹)((1st ‘𝐻)‘𝑥))‘(𝐴‘𝑥))))
Theorem	fucorid 50085*	Pre-composing a natural transformation with the identity natural transformation of a functor is pre-composing it with the object part of the functor, in maps-to notation. (Contributed by Zhi Wang, 11-Oct-2025.)
⊢ (𝜑 → (2nd ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸)) = 𝑃)    &   ⊢ 𝐼 = (Id‘𝑄)    &   ⊢ 𝑄 = (𝐶 FuncCat 𝐷)    &   ⊢ (𝜑 → 𝐴 ∈ (𝐺(𝐷 Nat 𝐸)𝐻))    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    ⇒   ⊢ (𝜑 → (𝐴(⟨𝐺, 𝐹⟩𝑃⟨𝐻, 𝐹⟩)(𝐼‘𝐹)) = (𝑥 ∈ (Base‘𝐶) ↦ (𝐴‘((1st ‘𝐹)‘𝑥))))
Theorem	fucorid2 50086	Pre-composing a natural transformation with the identity natural transformation of a functor is pre-composing it with the object part of the functor. (Contributed by Zhi Wang, 11-Oct-2025.)
⊢ (𝜑 → (2nd ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸)) = 𝑃)    &   ⊢ 𝐼 = (Id‘𝑄)    &   ⊢ 𝑄 = (𝐶 FuncCat 𝐷)    &   ⊢ (𝜑 → 𝐴 ∈ (𝐺(𝐷 Nat 𝐸)𝐻))    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    ⇒   ⊢ (𝜑 → (𝐴(⟨𝐺, 𝐹⟩𝑃⟨𝐻, 𝐹⟩)(𝐼‘𝐹)) = (𝐴 ∘ (1st ‘𝐹)))
21.50.15.19  Post-composition functors
Theorem	postcofval 50087*	Value of the post-composition functor as a curry of the functor composition bifunctor. (Contributed by Zhi Wang, 11-Oct-2025.)
⊢ 𝑄 = (𝐶 FuncCat 𝐷)    &   ⊢ 𝑅 = (𝐷 FuncCat 𝐸)    &   ⊢ ⚬ = (⟨𝑅, 𝑄⟩ curryF (⟨𝐶, 𝐷⟩ ∘F 𝐸))    &   ⊢ (𝜑 → 𝐹 ∈ (𝐷 Func 𝐸))    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ 𝐾 = ((1st ‘ ⚬ )‘𝐹)    ⇒   ⊢ (𝜑 → 𝐾 = ⟨(𝑔 ∈ (𝐶 Func 𝐷) ↦ (𝐹 ∘func 𝑔)), (𝑔 ∈ (𝐶 Func 𝐷), ℎ ∈ (𝐶 Func 𝐷) ↦ (𝑎 ∈ (𝑔(𝐶 Nat 𝐷)ℎ) ↦ (𝑥 ∈ (Base‘𝐶) ↦ ((((1st ‘𝑔)‘𝑥)(2nd ‘𝐹)((1st ‘ℎ)‘𝑥))‘(𝑎‘𝑥)))))⟩)
Theorem	postcofcl 50088	The post-composition functor as a curry of the functor composition bifunctor is a functor. (Contributed by Zhi Wang, 11-Oct-2025.)
⊢ 𝑄 = (𝐶 FuncCat 𝐷)    &   ⊢ 𝑅 = (𝐷 FuncCat 𝐸)    &   ⊢ ⚬ = (⟨𝑅, 𝑄⟩ curryF (⟨𝐶, 𝐷⟩ ∘F 𝐸))    &   ⊢ (𝜑 → 𝐹 ∈ (𝐷 Func 𝐸))    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ 𝐾 = ((1st ‘ ⚬ )‘𝐹)    &   ⊢ 𝑆 = (𝐶 FuncCat 𝐸)    ⇒   ⊢ (𝜑 → 𝐾 ∈ (𝑄 Func 𝑆))
21.50.15.20  Pre-composition functors
Theorem	precofvallem 50089	Lemma for precofval 50090 to enable catlid 17738 or catrid 17739. (Contributed by Zhi Wang, 11-Oct-2025.)
⊢ 𝐴 = (Base‘𝐶)    &   ⊢ 𝐵 = (Base‘𝐸)    &   ⊢ 1 = (Id‘𝐷)    &   ⊢ 𝐼 = (Id‘𝐸)    &   ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)    &   ⊢ (𝜑 → 𝐾(𝐷 Func 𝐸)𝐿)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    ⇒   ⊢ (𝜑 → ((((𝐹‘𝑋)𝐿(𝐹‘𝑋))‘(( 1 ∘ 𝐹)‘𝑋)) = (𝐼‘(𝐾‘(𝐹‘𝑋))) ∧ (𝐾‘(𝐹‘𝑋)) ∈ 𝐵))
Theorem	precofval 50090*	Value of the pre-composition functor as a transposed curry of the functor composition bifunctor. (Contributed by Zhi Wang, 11-Oct-2025.)
⊢ 𝑄 = (𝐶 FuncCat 𝐷)    &   ⊢ 𝑅 = (𝐷 FuncCat 𝐸)    &   ⊢ (𝜑 → ⚬ = (⟨𝑄, 𝑅⟩ curryF ((⟨𝐶, 𝐷⟩ ∘F 𝐸) ∘func (𝑄 swapF 𝑅))))    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    &   ⊢ (𝜑 → 𝐸 ∈ Cat)    &   ⊢ (𝜑 → 𝐾 = ((1st ‘ ⚬ )‘𝐹))    ⇒   ⊢ (𝜑 → 𝐾 = ⟨(𝑔 ∈ (𝐷 Func 𝐸) ↦ (𝑔 ∘func 𝐹)), (𝑔 ∈ (𝐷 Func 𝐸), ℎ ∈ (𝐷 Func 𝐸) ↦ (𝑎 ∈ (𝑔(𝐷 Nat 𝐸)ℎ) ↦ (𝑥 ∈ (Base‘𝐶) ↦ (𝑎‘((1st ‘𝐹)‘𝑥)))))⟩)
Theorem	precofvalALT 50091*	Alternate proof of precofval 50090. (Contributed by Zhi Wang, 11-Oct-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝑄 = (𝐶 FuncCat 𝐷)    &   ⊢ 𝑅 = (𝐷 FuncCat 𝐸)    &   ⊢ (𝜑 → ⚬ = (⟨𝑄, 𝑅⟩ curryF ((⟨𝐶, 𝐷⟩ ∘F 𝐸) ∘func (𝑄 swapF 𝑅))))    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    &   ⊢ (𝜑 → 𝐸 ∈ Cat)    &   ⊢ (𝜑 → 𝐾 = ((1st ‘ ⚬ )‘𝐹))    ⇒   ⊢ (𝜑 → 𝐾 = ⟨(𝑔 ∈ (𝐷 Func 𝐸) ↦ (𝑔 ∘func 𝐹)), (𝑔 ∈ (𝐷 Func 𝐸), ℎ ∈ (𝐷 Func 𝐸) ↦ (𝑎 ∈ (𝑔(𝐷 Nat 𝐸)ℎ) ↦ (𝑥 ∈ (Base‘𝐶) ↦ (𝑎‘((1st ‘𝐹)‘𝑥)))))⟩)
Theorem	precofval2 50092*	Value of the pre-composition functor as a transposed curry of the functor composition bifunctor. (Contributed by Zhi Wang, 11-Oct-2025.)
⊢ 𝑄 = (𝐶 FuncCat 𝐷)    &   ⊢ 𝑅 = (𝐷 FuncCat 𝐸)    &   ⊢ (𝜑 → ⚬ = (⟨𝑄, 𝑅⟩ curryF ((⟨𝐶, 𝐷⟩ ∘F 𝐸) ∘func (𝑄 swapF 𝑅))))    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    &   ⊢ (𝜑 → 𝐸 ∈ Cat)    &   ⊢ (𝜑 → 𝐾 = ((1st ‘ ⚬ )‘𝐹))    ⇒   ⊢ (𝜑 → 𝐾 = ⟨(𝑔 ∈ (𝐷 Func 𝐸) ↦ (𝑔 ∘func 𝐹)), (𝑔 ∈ (𝐷 Func 𝐸), ℎ ∈ (𝐷 Func 𝐸) ↦ (𝑎 ∈ (𝑔(𝐷 Nat 𝐸)ℎ) ↦ (𝑎 ∘ (1st ‘𝐹))))⟩)
Theorem	precofcl 50093	The pre-composition functor as a transposed curry of the functor composition bifunctor is a functor. (Contributed by Zhi Wang, 11-Oct-2025.)
⊢ 𝑄 = (𝐶 FuncCat 𝐷)    &   ⊢ 𝑅 = (𝐷 FuncCat 𝐸)    &   ⊢ (𝜑 → ⚬ = (⟨𝑄, 𝑅⟩ curryF ((⟨𝐶, 𝐷⟩ ∘F 𝐸) ∘func (𝑄 swapF 𝑅))))    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    &   ⊢ (𝜑 → 𝐸 ∈ Cat)    &   ⊢ (𝜑 → 𝐾 = ((1st ‘ ⚬ )‘𝐹))    &   ⊢ 𝑆 = (𝐶 FuncCat 𝐸)    ⇒   ⊢ (𝜑 → 𝐾 ∈ (𝑅 Func 𝑆))
Theorem	precofval3 50094*	Value of the pre-composition functor as a transposed curry of the functor composition bifunctor. (Contributed by Zhi Wang, 20-Oct-2025.)
⊢ 𝑅 = (𝐷 FuncCat 𝐸)    &   ⊢ 𝐵 = (𝐷 Func 𝐸)    &   ⊢ 𝑁 = (𝐷 Nat 𝐸)    &   ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)    &   ⊢ (𝜑 → 𝐸 ∈ Cat)    &   ⊢ (𝜑 → 𝐾 = (𝑔 ∈ 𝐵 ↦ (𝑔 ∘func ⟨𝐹, 𝐺⟩)))    &   ⊢ (𝜑 → 𝐿 = (𝑔 ∈ 𝐵, ℎ ∈ 𝐵 ↦ (𝑎 ∈ (𝑔𝑁ℎ) ↦ (𝑎 ∘ 𝐹))))    &   ⊢ 𝑄 = (𝐶 FuncCat 𝐷)    &   ⊢ (𝜑 → ⚬ = (⟨𝑄, 𝑅⟩ curryF ((⟨𝐶, 𝐷⟩ ∘F 𝐸) ∘func (𝑄 swapF 𝑅))))    &   ⊢ (𝜑 → 𝑀 = ((1st ‘ ⚬ )‘⟨𝐹, 𝐺⟩))    ⇒   ⊢ (𝜑 → ⟨𝐾, 𝐿⟩ = 𝑀)
Theorem	precoffunc 50095*	The pre-composition functor, expressed explicitly, is a functor. (Contributed by Zhi Wang, 11-Oct-2025.) (Proof shortened by Zhi Wang, 20-Oct-2025.)
⊢ 𝑅 = (𝐷 FuncCat 𝐸)    &   ⊢ 𝐵 = (𝐷 Func 𝐸)    &   ⊢ 𝑁 = (𝐷 Nat 𝐸)    &   ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)    &   ⊢ (𝜑 → 𝐸 ∈ Cat)    &   ⊢ (𝜑 → 𝐾 = (𝑔 ∈ 𝐵 ↦ (𝑔 ∘func ⟨𝐹, 𝐺⟩)))    &   ⊢ (𝜑 → 𝐿 = (𝑔 ∈ 𝐵, ℎ ∈ 𝐵 ↦ (𝑎 ∈ (𝑔𝑁ℎ) ↦ (𝑎 ∘ 𝐹))))    &   ⊢ 𝑆 = (𝐶 FuncCat 𝐸)    ⇒   ⊢ (𝜑 → 𝐾(𝑅 Func 𝑆)𝐿)
Syntax	cprcof 50096	Extend class notation with pre-composition functors.
class −∘F
Definition	df-prcof 50097*	Definition of pre-composition functors. The object part of the pre-composition functor given by 𝐹 pre-composes a functor with 𝐹; the morphism part pre-composes a natural transformation with the object part of 𝐹, in terms of function composition. Comments before the definition in § 3 of Chapter X in p. 236 of Mac Lane, Saunders, Categories for the Working Mathematician, 2nd Edition, Springer Science+Business Media, New York, (1998) [QA169.M33 1998]; available at https://math.mit.edu/~hrm/palestine/maclane-categories.pdf (retrieved 3 Nov 2025). The notation −∘F is inspired by this page: https://1lab.dev/Cat.Functor.Compose.html. The pre-composition functor can also be defined as a transposed curry of the functor composition bifunctor (precofval3 50094). But such definition requires an explicit third category. prcoftposcurfuco 50106 and prcoftposcurfucoa 50107 prove the equivalence. (Contributed by Zhi Wang, 2-Nov-2025.)
⊢ −∘F = (𝑝 ∈ V, 𝑓 ∈ V ↦ ⦋(1st ‘𝑝) / 𝑑⦌⦋(2nd ‘𝑝) / 𝑒⦌⦋(𝑑 Func 𝑒) / 𝑏⦌⟨(𝑘 ∈ 𝑏 ↦ (𝑘 ∘func 𝑓)), (𝑘 ∈ 𝑏, 𝑙 ∈ 𝑏 ↦ (𝑎 ∈ (𝑘(𝑑 Nat 𝑒)𝑙) ↦ (𝑎 ∘ (1st ‘𝑓))))⟩)
Theorem	reldmprcof 50098	The domain of −∘F is a relation. (Contributed by Zhi Wang, 2-Nov-2025.)
⊢ Rel dom −∘F
Theorem	prcofvalg 50099*	Value of the pre-composition functor. (Contributed by Zhi Wang, 2-Nov-2025.)
⊢ 𝐵 = (𝐷 Func 𝐸)    &   ⊢ 𝑁 = (𝐷 Nat 𝐸)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑈)    &   ⊢ (𝜑 → 𝑃 ∈ 𝑉)    &   ⊢ (𝜑 → (1st ‘𝑃) = 𝐷)    &   ⊢ (𝜑 → (2nd ‘𝑃) = 𝐸)    ⇒   ⊢ (𝜑 → (𝑃 −∘F 𝐹) = ⟨(𝑘 ∈ 𝐵 ↦ (𝑘 ∘func 𝐹)), (𝑘 ∈ 𝐵, 𝑙 ∈ 𝐵 ↦ (𝑎 ∈ (𝑘𝑁𝑙) ↦ (𝑎 ∘ (1st ‘𝐹))))⟩)
Theorem	prcofvala 50100*	Value of the pre-composition functor. (Contributed by Zhi Wang, 2-Nov-2025.)
⊢ 𝐵 = (𝐷 Func 𝐸)    &   ⊢ 𝑁 = (𝐷 Nat 𝐸)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑈)    ⇒   ⊢ (𝜑 → (⟨𝐷, 𝐸⟩ −∘F 𝐹) = ⟨(𝑘 ∈ 𝐵 ↦ (𝑘 ∘func 𝐹)), (𝑘 ∈ 𝐵, 𝑙 ∈ 𝐵 ↦ (𝑎 ∈ (𝑘𝑁𝑙) ↦ (𝑎 ∘ (1st ‘𝐹))))⟩)