P. 498 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 49701-49800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	fuco22natlem1 49701	Lemma for fuco22nat 49705. 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 49702	Lemma for fuco22nat 49705. The commutative square of natural transformation 𝐵 in category 𝐸, combined with the commutative square of fuco22natlem1 49701. (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 49703	Combine fuco22natlem2 49702 with fuco23 49700. (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 49704	The composed natural transformation is a natural transformation. Use fuco22nat 49705 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 49705	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 49706	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 49707	Each identity morphism in the source category is mapped to the corresponding identity morphism in the target category. See also fucoid2 49708. (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 49708	Each identity morphism in the source category is mapped to the corresponding identity morphism in the target category. See also fucoid 49707. (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 49709*	The morphism part of the functor composition bifunctor. See also fuco22 49698. (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 49710	The naturality property (nati 17894) 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 49711	The morphism part of the functor composition bifunctor. An alternate definition of ∘F. See also fuco23 49700. (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 49712	Lemma for fucoco 49716. 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 49713*	Lemma for fucoco 49716. 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 49714*	Lemma for fucoco 49716. 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 49715*	Lemma for fucoco 49716. 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 49716	Composition in the source category is mapped to composition in the target. See also fucoco2 49717. (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 49717	Composition in the source category is mapped to composition in the target. See also fucoco 49716. (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 49718	The functor composition bifunctor is a functor. See also fucofunca 49719. 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 49832), for example, (SetCat‘1o) (the trivial category, setc1oterm 49850), 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 49700. 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 49676 up to fuco23 49700, but ((𝐵(𝑈𝑃𝑉)𝐴)‘𝑋) might be mapped to a different morphism in category 𝐸. See fucofulem2 49670 for some insights. 2. fuco22nat 49705, fucoid 49707, and fucoco 49716 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 49719	The functor composition bifunctor is a functor. See also fucofunc 49718. (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 49720*	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 49721*	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 49722	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)))
21.49.15.19  Post-composition functors
Theorem	postcofval 49723*	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 49724	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))
21.49.15.20  Pre-composition functors
Theorem	precofvallem 49725	Lemma for precofval 49726 to enable catlid 17618 or catrid 17619. (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 49726*	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 49727*	Alternate proof of precofval 49726. (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 49728*	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 49729	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 49730*	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 49731*	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 49732	Extend class notation with pre-composition functors.
class −∘F
Definition	df-prcof 49733*	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 49730). But such definition requires an explicit third category. prcoftposcurfuco 49742 and prcoftposcurfucoa 49743 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 49734	The domain of −∘F is a relation. (Contributed by Zhi Wang, 2-Nov-2025.)
⊢ Rel dom −∘F
Theorem	prcofvalg 49735*	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 49736*	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))))⟩)
Theorem	prcofval 49737*	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)    &   ⊢ Rel 𝑅    &   ⊢ (𝜑 → 𝐹𝑅𝐺)    ⇒   ⊢ (𝜑 → (⟨𝐷, 𝐸⟩ −∘F ⟨𝐹, 𝐺⟩) = ⟨(𝑘 ∈ 𝐵 ↦ (𝑘 ∘func ⟨𝐹, 𝐺⟩)), (𝑘 ∈ 𝐵, 𝑙 ∈ 𝐵 ↦ (𝑎 ∈ (𝑘𝑁𝑙) ↦ (𝑎 ∘ 𝐹)))⟩)
Theorem	prcofpropd 49738	If the categories have the same set of objects, morphisms, and compositions, then they have the same pre-composition functors. (Contributed by Zhi Wang, 21-Nov-2025.)
⊢ (𝜑 → (Homf ‘𝐴) = (Homf ‘𝐵))    &   ⊢ (𝜑 → (compf‘𝐴) = (compf‘𝐵))    &   ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))    &   ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑊)    ⇒   ⊢ (𝜑 → (⟨𝐴, 𝐶⟩ −∘F 𝐹) = (⟨𝐵, 𝐷⟩ −∘F 𝐹))
Theorem	prcofelvv 49739	The pre-composition functor is an ordered pair. (Contributed by Zhi Wang, 4-Nov-2025.)
⊢ (𝜑 → 𝐹 ∈ 𝑈)    &   ⊢ (𝜑 → 𝑃 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝑃 −∘F 𝐹) ∈ (V × V))
Theorem	reldmprcof1 49740	The domain of the object part of the pre-composition functor is a relation. (Contributed by Zhi Wang, 2-Nov-2025.)
⊢ Rel dom (1st ‘(𝑃 −∘F 𝐹))
Theorem	reldmprcof2 49741	The domain of the morphism part of the pre-composition functor is a relation. (Contributed by Zhi Wang, 2-Nov-2025.)
⊢ Rel dom (2nd ‘(𝑃 −∘F 𝐹))
Theorem	prcoftposcurfuco 49742	The pre-composition functor is the transposed curry of the functor composition bifunctor. (Contributed by Zhi Wang, 2-Nov-2025.)
⊢ 𝑅 = (𝐷 FuncCat 𝐸)    &   ⊢ (𝜑 → 𝐸 ∈ Cat)    &   ⊢ 𝑄 = (𝐶 FuncCat 𝐷)    &   ⊢ (𝜑 → ⚬ = (⟨𝑄, 𝑅⟩ curryF ((⟨𝐶, 𝐷⟩ ∘F 𝐸) ∘func (𝑄 swapF 𝑅))))    &   ⊢ (𝜑 → 𝑀 = ((1st ‘ ⚬ )‘⟨𝐹, 𝐺⟩))    &   ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)    ⇒   ⊢ (𝜑 → (⟨𝐷, 𝐸⟩ −∘F ⟨𝐹, 𝐺⟩) = 𝑀)
Theorem	prcoftposcurfucoa 49743	The pre-composition functor is the transposed curry of the functor composition bifunctor. (Contributed by Zhi Wang, 2-Nov-2025.)
⊢ 𝑅 = (𝐷 FuncCat 𝐸)    &   ⊢ (𝜑 → 𝐸 ∈ Cat)    &   ⊢ 𝑄 = (𝐶 FuncCat 𝐷)    &   ⊢ (𝜑 → ⚬ = (⟨𝑄, 𝑅⟩ curryF ((⟨𝐶, 𝐷⟩ ∘F 𝐸) ∘func (𝑄 swapF 𝑅))))    &   ⊢ (𝜑 → 𝑀 = ((1st ‘ ⚬ )‘𝐹))    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    ⇒   ⊢ (𝜑 → (⟨𝐷, 𝐸⟩ −∘F 𝐹) = 𝑀)
Theorem	prcoffunc 49744	The pre-composition functor is a functor. (Contributed by Zhi Wang, 2-Nov-2025.)
⊢ 𝑅 = (𝐷 FuncCat 𝐸)    &   ⊢ (𝜑 → 𝐸 ∈ Cat)    &   ⊢ 𝑆 = (𝐶 FuncCat 𝐸)    &   ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)    ⇒   ⊢ (𝜑 → (⟨𝐷, 𝐸⟩ −∘F ⟨𝐹, 𝐺⟩) ∈ (𝑅 Func 𝑆))
Theorem	prcoffunca 49745	The pre-composition functor is a functor. (Contributed by Zhi Wang, 2-Nov-2025.)
⊢ 𝑅 = (𝐷 FuncCat 𝐸)    &   ⊢ (𝜑 → 𝐸 ∈ Cat)    &   ⊢ 𝑆 = (𝐶 FuncCat 𝐸)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    ⇒   ⊢ (𝜑 → (⟨𝐷, 𝐸⟩ −∘F 𝐹) ∈ (𝑅 Func 𝑆))
Theorem	prcoffunca2 49746	The pre-composition functor is a functor. (Contributed by Zhi Wang, 4-Nov-2025.)
⊢ 𝑅 = (𝐷 FuncCat 𝐸)    &   ⊢ (𝜑 → 𝐸 ∈ Cat)    &   ⊢ 𝑆 = (𝐶 FuncCat 𝐸)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    &   ⊢ (𝜑 → (⟨𝐷, 𝐸⟩ −∘F 𝐹) = ⟨𝐾, 𝐿⟩)    ⇒   ⊢ (𝜑 → 𝐾(𝑅 Func 𝑆)𝐿)
Theorem	prcof1 49747	The object part of the pre-composition functor. (Contributed by Zhi Wang, 3-Nov-2025.)
⊢ (𝜑 → 𝐾 ∈ (𝐷 Func 𝐸))    &   ⊢ (𝜑 → (1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)) = 𝑂)    ⇒   ⊢ (𝜑 → (𝑂‘𝐾) = (𝐾 ∘func 𝐹))
Theorem	prcof2a 49748*	The morphism part of the pre-composition functor. (Contributed by Zhi Wang, 3-Nov-2025.)
⊢ 𝑁 = (𝐷 Nat 𝐸)    &   ⊢ (𝜑 → 𝐾 ∈ (𝐷 Func 𝐸))    &   ⊢ (𝜑 → 𝐿 ∈ (𝐷 Func 𝐸))    &   ⊢ (𝜑 → (2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)) = 𝑃)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑈)    ⇒   ⊢ (𝜑 → (𝐾𝑃𝐿) = (𝑎 ∈ (𝐾𝑁𝐿) ↦ (𝑎 ∘ (1st ‘𝐹))))
Theorem	prcof2 49749*	The morphism part of the pre-composition functor. (Contributed by Zhi Wang, 3-Nov-2025.)
⊢ 𝑁 = (𝐷 Nat 𝐸)    &   ⊢ (𝜑 → 𝐾 ∈ (𝐷 Func 𝐸))    &   ⊢ (𝜑 → 𝐿 ∈ (𝐷 Func 𝐸))    &   ⊢ (𝜑 → (2nd ‘(⟨𝐷, 𝐸⟩ −∘F ⟨𝐹, 𝐺⟩)) = 𝑃)    &   ⊢ Rel 𝑅    &   ⊢ (𝜑 → 𝐹𝑅𝐺)    ⇒   ⊢ (𝜑 → (𝐾𝑃𝐿) = (𝑎 ∈ (𝐾𝑁𝐿) ↦ (𝑎 ∘ 𝐹)))
Theorem	prcof21a 49750	The morphism part of the pre-composition functor. (Contributed by Zhi Wang, 3-Nov-2025.)
⊢ 𝑁 = (𝐷 Nat 𝐸)    &   ⊢ (𝜑 → 𝐴 ∈ (𝐾𝑁𝐿))    &   ⊢ (𝜑 → (2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)) = 𝑃)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑈)    ⇒   ⊢ (𝜑 → ((𝐾𝑃𝐿)‘𝐴) = (𝐴 ∘ (1st ‘𝐹)))
Theorem	prcof22a 49751	The morphism part of the pre-composition functor. (Contributed by Zhi Wang, 3-Nov-2025.)
⊢ 𝑁 = (𝐷 Nat 𝐸)    &   ⊢ (𝜑 → 𝐴 ∈ (𝐾𝑁𝐿))    &   ⊢ (𝜑 → (2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)) = 𝑃)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    ⇒   ⊢ (𝜑 → (((𝐾𝑃𝐿)‘𝐴)‘𝑋) = (𝐴‘((1st ‘𝐹)‘𝑋)))
Theorem	prcofdiag1 49752	A constant functor pre-composed by a functor is another constant functor. (Contributed by Zhi Wang, 25-Nov-2025.)
⊢ 𝐿 = (𝐶Δfunc𝐷)    &   ⊢ 𝑀 = (𝐶Δfunc𝐸)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐸 Func 𝐷))    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (((1st ‘𝐿)‘𝑋) ∘func 𝐹) = ((1st ‘𝑀)‘𝑋))
Theorem	prcofdiag 49753	A diagonal functor post-composed by a pre-composition functor is another diagonal functor. (Contributed by Zhi Wang, 25-Nov-2025.)
⊢ 𝐿 = (𝐶Δfunc𝐷)    &   ⊢ 𝑀 = (𝐶Δfunc𝐸)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐸 Func 𝐷))    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → (⟨𝐷, 𝐶⟩ −∘F 𝐹) = 𝐺)    ⇒   ⊢ (𝜑 → (𝐺 ∘func 𝐿) = 𝑀)
21.49.16  Examples of categories
21.49.16.1  The category of categories
Theorem	catcrcl 49754	Reverse closure for the category of categories (in a universe) (Contributed by Zhi Wang, 14-Nov-2025.)
⊢ 𝐶 = (CatCat‘𝑈)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))    ⇒   ⊢ (𝜑 → 𝑈 ∈ V)
Theorem	catcrcl2 49755	Reverse closure for the category of categories (in a universe) (Contributed by Zhi Wang, 14-Nov-2025.)
⊢ 𝐶 = (CatCat‘𝑈)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))    &   ⊢ 𝐵 = (Base‘𝐶)    ⇒   ⊢ (𝜑 → (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵))
Theorem	elcatchom 49756	A morphism of the category of categories (in a universe) is a functor. See df-catc 18035 for the definition of the category Cat, which consists of all categories in the universe 𝑢 (i.e., "𝑢-small categories", see Definition 3.44. of [Adamek] p. 39), with functors as the morphisms (catchom 18039). (Contributed by Zhi Wang, 14-Nov-2025.)
⊢ 𝐶 = (CatCat‘𝑈)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))    ⇒   ⊢ (𝜑 → 𝐹 ∈ (𝑋 Func 𝑌))
Theorem	catcsect 49757	The property "𝐹 is a section of 𝐺 " in a category of small categories (in a universe). (Contributed by Zhi Wang, 14-Nov-2025.)
⊢ 𝐶 = (CatCat‘𝑈)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ 𝐼 = (idfunc‘𝑋)    &   ⊢ 𝑆 = (Sect‘𝐶)    ⇒   ⊢ (𝐹(𝑋𝑆𝑌)𝐺 ↔ ((𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝐺 ∈ (𝑌𝐻𝑋)) ∧ (𝐺 ∘func 𝐹) = 𝐼))
Theorem	catcinv 49758	The property "𝐹 is an inverse of 𝐺 " in a category of small categories (in a universe). (Contributed by Zhi Wang, 14-Nov-2025.)
⊢ 𝐶 = (CatCat‘𝑈)    &   ⊢ 𝑁 = (Inv‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ 𝐼 = (idfunc‘𝑋)    &   ⊢ 𝐽 = (idfunc‘𝑌)    ⇒   ⊢ (𝐹(𝑋𝑁𝑌)𝐺 ↔ ((𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝐺 ∈ (𝑌𝐻𝑋)) ∧ ((𝐺 ∘func 𝐹) = 𝐼 ∧ (𝐹 ∘func 𝐺) = 𝐽)))
Theorem	catcisoi 49759	A functor is an isomorphism of categories only if it is full and faithful, and is a bijection on the objects. Remark 3.28(2) in [Adamek] p. 34. (Contributed by Zhi Wang, 17-Nov-2025.)
⊢ 𝐶 = (CatCat‘𝑈)    &   ⊢ 𝑅 = (Base‘𝑋)    &   ⊢ 𝑆 = (Base‘𝑌)    &   ⊢ 𝐼 = (Iso‘𝐶)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐼𝑌))    ⇒   ⊢ (𝜑 → (𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1st ‘𝐹):𝑅–1-1-onto→𝑆))
Theorem	uobeq2 49760	If a full functor (in fact, a full embedding) is a section, then the sets of universal objects are equal. (Contributed by Zhi Wang, 17-Nov-2025.)
⊢ 𝐵 = (Base‘𝐷)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    &   ⊢ (𝜑 → (𝐾 ∘func 𝐹) = 𝐺)    &   ⊢ (𝜑 → ((1st ‘𝐾)‘𝑋) = 𝑌)    &   ⊢ 𝑄 = (CatCat‘𝑈)    &   ⊢ 𝑆 = (Sect‘𝑄)    &   ⊢ (𝜑 → 𝐾 ∈ (𝐷 Full 𝐸))    &   ⊢ (𝜑 → 𝐾 ∈ dom (𝐷𝑆𝐸))    ⇒   ⊢ (𝜑 → dom (𝐹(𝐶 UP 𝐷)𝑋) = dom (𝐺(𝐶 UP 𝐸)𝑌))
Theorem	uobeq3 49761	An isomorphism between categories generates equal sets of universal objects. (Contributed by Zhi Wang, 17-Nov-2025.)
⊢ 𝐵 = (Base‘𝐷)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    &   ⊢ (𝜑 → (𝐾 ∘func 𝐹) = 𝐺)    &   ⊢ (𝜑 → ((1st ‘𝐾)‘𝑋) = 𝑌)    &   ⊢ 𝑄 = (CatCat‘𝑈)    &   ⊢ 𝐼 = (Iso‘𝑄)    &   ⊢ (𝜑 → 𝐾 ∈ (𝐷𝐼𝐸))    ⇒   ⊢ (𝜑 → dom (𝐹(𝐶 UP 𝐷)𝑋) = dom (𝐺(𝐶 UP 𝐸)𝑌))
Theorem	opf11 49762	The object part of the op functor on functor categories. Lemma for fucoppc 49769. (Contributed by Zhi Wang, 18-Nov-2025.)
⊢ (𝜑 → 𝐹 = ( oppFunc ↾ (𝐶 Func 𝐷)))    &   ⊢ (𝜑 → 𝑋 ∈ (𝐶 Func 𝐷))    ⇒   ⊢ (𝜑 → (1st ‘(𝐹‘𝑋)) = (1st ‘𝑋))
Theorem	opf12 49763	The object part of the op functor on functor categories. Lemma for oppfdiag 49775. (Contributed by Zhi Wang, 19-Nov-2025.)
⊢ (𝜑 → 𝐹 = ( oppFunc ↾ (𝐶 Func 𝐷)))    &   ⊢ (𝜑 → 𝑋 ∈ (𝐶 Func 𝐷))    ⇒   ⊢ (𝜑 → (𝑀(2nd ‘(𝐹‘𝑋))𝑁) = (𝑁(2nd ‘𝑋)𝑀))
Theorem	opf2fval 49764*	The morphism part of the op functor on functor categories. Lemma for fucoppc 49769. (Contributed by Zhi Wang, 18-Nov-2025.)
⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑦𝑁𝑥))))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑋𝐹𝑌) = ( I ↾ (𝑌𝑁𝑋)))
Theorem	opf2 49765*	The morphism part of the op functor on functor categories. Lemma for fucoppc 49769. (Contributed by Zhi Wang, 18-Nov-2025.)
⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑦𝑁𝑥))))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    &   ⊢ (𝜑 → 𝐷 ∈ (𝑌𝑁𝑋))    ⇒   ⊢ (𝜑 → ((𝑋𝐹𝑌)‘𝐶) = 𝐷)
Theorem	fucoppclem 49766	Lemma for fucoppc 49769. (Contributed by Zhi Wang, 18-Nov-2025.)
⊢ 𝑂 = (oppCat‘𝐶)    &   ⊢ 𝑃 = (oppCat‘𝐷)    &   ⊢ 𝑁 = (𝐶 Nat 𝐷)    &   ⊢ (𝜑 → 𝐹 = ( oppFunc ↾ (𝐶 Func 𝐷)))    &   ⊢ (𝜑 → 𝑋 ∈ (𝐶 Func 𝐷))    &   ⊢ (𝜑 → 𝑌 ∈ (𝐶 Func 𝐷))    ⇒   ⊢ (𝜑 → (𝑌𝑁𝑋) = ((𝐹‘𝑋)(𝑂 Nat 𝑃)(𝐹‘𝑌)))
Theorem	fucoppcid 49767*	The opposite category of functors is compatible with the category of opposite functors in terms of identity morphism. (Contributed by Zhi Wang, 18-Nov-2025.)
⊢ 𝑂 = (oppCat‘𝐶)    &   ⊢ 𝑃 = (oppCat‘𝐷)    &   ⊢ 𝑄 = (𝐶 FuncCat 𝐷)    &   ⊢ 𝑅 = (oppCat‘𝑄)    &   ⊢ 𝑆 = (𝑂 FuncCat 𝑃)    &   ⊢ 𝑁 = (𝐶 Nat 𝐷)    &   ⊢ (𝜑 → 𝐹 = ( oppFunc ↾ (𝐶 Func 𝐷)))    &   ⊢ (𝜑 → 𝐺 = (𝑥 ∈ (𝐶 Func 𝐷), 𝑦 ∈ (𝐶 Func 𝐷) ↦ ( I ↾ (𝑦𝑁𝑥))))    &   ⊢ (𝜑 → 𝑋 ∈ (𝐶 Func 𝐷))    ⇒   ⊢ (𝜑 → ((𝑋𝐺𝑋)‘((Id‘𝑅)‘𝑋)) = ((Id‘𝑆)‘(𝐹‘𝑋)))
Theorem	fucoppcco 49768*	The opposite category of functors is compatible with the category of opposite functors in terms of composition. (Contributed by Zhi Wang, 18-Nov-2025.)
⊢ 𝑂 = (oppCat‘𝐶)    &   ⊢ 𝑃 = (oppCat‘𝐷)    &   ⊢ 𝑄 = (𝐶 FuncCat 𝐷)    &   ⊢ 𝑅 = (oppCat‘𝑄)    &   ⊢ 𝑆 = (𝑂 FuncCat 𝑃)    &   ⊢ 𝑁 = (𝐶 Nat 𝐷)    &   ⊢ (𝜑 → 𝐹 = ( oppFunc ↾ (𝐶 Func 𝐷)))    &   ⊢ (𝜑 → 𝐺 = (𝑥 ∈ (𝐶 Func 𝐷), 𝑦 ∈ (𝐶 Func 𝐷) ↦ ( I ↾ (𝑦𝑁𝑥))))    &   ⊢ (𝜑 → 𝐴 ∈ (𝑋(Hom ‘𝑅)𝑌))    &   ⊢ (𝜑 → 𝐵 ∈ (𝑌(Hom ‘𝑅)𝑍))    ⇒   ⊢ (𝜑 → ((𝑋𝐺𝑍)‘(𝐵(⟨𝑋, 𝑌⟩(comp‘𝑅)𝑍)𝐴)) = (((𝑌𝐺𝑍)‘𝐵)(⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩(comp‘𝑆)(𝐹‘𝑍))((𝑋𝐺𝑌)‘𝐴)))
Theorem	fucoppc 49769*	The isomorphism from the opposite category of functors to the category of opposite functors. (Contributed by Zhi Wang, 18-Nov-2025.)
⊢ 𝑂 = (oppCat‘𝐶)    &   ⊢ 𝑃 = (oppCat‘𝐷)    &   ⊢ 𝑄 = (𝐶 FuncCat 𝐷)    &   ⊢ 𝑅 = (oppCat‘𝑄)    &   ⊢ 𝑆 = (𝑂 FuncCat 𝑃)    &   ⊢ 𝑁 = (𝐶 Nat 𝐷)    &   ⊢ (𝜑 → 𝐹 = ( oppFunc ↾ (𝐶 Func 𝐷)))    &   ⊢ (𝜑 → 𝐺 = (𝑥 ∈ (𝐶 Func 𝐷), 𝑦 ∈ (𝐶 Func 𝐷) ↦ ( I ↾ (𝑦𝑁𝑥))))    &   ⊢ 𝑇 = (CatCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑇)    &   ⊢ 𝐼 = (Iso‘𝑇)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑅 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑆 ∈ 𝐵)    ⇒   ⊢ (𝜑 → 𝐹(𝑅𝐼𝑆)𝐺)
Theorem	fucoppcffth 49770*	A fully faithful functor from the opposite category of functors to the category of opposite functors. (Contributed by Zhi Wang, 19-Nov-2025.)
⊢ 𝑂 = (oppCat‘𝐶)    &   ⊢ 𝑃 = (oppCat‘𝐷)    &   ⊢ 𝑄 = (𝐶 FuncCat 𝐷)    &   ⊢ 𝑅 = (oppCat‘𝑄)    &   ⊢ 𝑆 = (𝑂 FuncCat 𝑃)    &   ⊢ 𝑁 = (𝐶 Nat 𝐷)    &   ⊢ (𝜑 → 𝐹 = ( oppFunc ↾ (𝐶 Func 𝐷)))    &   ⊢ (𝜑 → 𝐺 = (𝑥 ∈ (𝐶 Func 𝐷), 𝑦 ∈ (𝐶 Func 𝐷) ↦ ( I ↾ (𝑦𝑁𝑥))))    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    ⇒   ⊢ (𝜑 → 𝐹((𝑅 Full 𝑆) ∩ (𝑅 Faith 𝑆))𝐺)
Theorem	fucoppcfunc 49771*	A functor from the opposite category of functors to the category of opposite functors. (Contributed by Zhi Wang, 19-Nov-2025.)
⊢ 𝑂 = (oppCat‘𝐶)    &   ⊢ 𝑃 = (oppCat‘𝐷)    &   ⊢ 𝑄 = (𝐶 FuncCat 𝐷)    &   ⊢ 𝑅 = (oppCat‘𝑄)    &   ⊢ 𝑆 = (𝑂 FuncCat 𝑃)    &   ⊢ 𝑁 = (𝐶 Nat 𝐷)    &   ⊢ (𝜑 → 𝐹 = ( oppFunc ↾ (𝐶 Func 𝐷)))    &   ⊢ (𝜑 → 𝐺 = (𝑥 ∈ (𝐶 Func 𝐷), 𝑦 ∈ (𝐶 Func 𝐷) ↦ ( I ↾ (𝑦𝑁𝑥))))    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    ⇒   ⊢ (𝜑 → 𝐹(𝑅 Func 𝑆)𝐺)
Theorem	fucoppccic 49772	The opposite category of functors is isomorphic to the category of opposite functors. (Contributed by Zhi Wang, 18-Nov-2025.)
⊢ 𝐶 = (CatCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝑋 = (oppCat‘(𝐷 FuncCat 𝐸))    &   ⊢ 𝑌 = ((oppCat‘𝐷) FuncCat (oppCat‘𝐸))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑊)    ⇒   ⊢ (𝜑 → 𝑋( ≃𝑐 ‘𝐶)𝑌)
Theorem	oppfdiag1 49773	A constant functor for opposite categories is the opposite functor of the constant functor for original categories. (Contributed by Zhi Wang, 19-Nov-2025.)
⊢ 𝑂 = (oppCat‘𝐶)    &   ⊢ 𝑃 = (oppCat‘𝐷)    &   ⊢ 𝐿 = (𝐶Δfunc𝐷)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ (𝜑 → 𝐹 = ( oppFunc ↾ (𝐷 Func 𝐶)))    &   ⊢ 𝐴 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    ⇒   ⊢ (𝜑 → (𝐹‘((1st ‘𝐿)‘𝑋)) = ((1st ‘(𝑂Δfunc𝑃))‘𝑋))
Theorem	oppfdiag1a 49774	A constant functor for opposite categories is the opposite functor of the constant functor for original categories. (Contributed by Zhi Wang, 19-Nov-2025.)
⊢ 𝑂 = (oppCat‘𝐶)    &   ⊢ 𝑃 = (oppCat‘𝐷)    &   ⊢ 𝐿 = (𝐶Δfunc𝐷)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ 𝐴 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    ⇒   ⊢ (𝜑 → ( oppFunc ‘((1st ‘𝐿)‘𝑋)) = ((1st ‘(𝑂Δfunc𝑃))‘𝑋))
Theorem	oppfdiag 49775*	A diagonal functor for opposite categories is the opposite functor of the diagonal functor for original categories post-composed by an isomorphism (fucoppc 49769). (Contributed by Zhi Wang, 19-Nov-2025.)
⊢ 𝑂 = (oppCat‘𝐶)    &   ⊢ 𝑃 = (oppCat‘𝐷)    &   ⊢ 𝐿 = (𝐶Δfunc𝐷)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ (𝜑 → 𝐹 = ( oppFunc ↾ (𝐷 Func 𝐶)))    &   ⊢ 𝑁 = (𝐷 Nat 𝐶)    &   ⊢ (𝜑 → 𝐺 = (𝑚 ∈ (𝐷 Func 𝐶), 𝑛 ∈ (𝐷 Func 𝐶) ↦ ( I ↾ (𝑛𝑁𝑚))))    ⇒   ⊢ (𝜑 → (⟨𝐹, 𝐺⟩ ∘func ( oppFunc ‘𝐿)) = (𝑂Δfunc𝑃))
21.49.16.2  Thin categories
Syntax	cthinc 49776	Extend class notation with the class of thin categories.
class ThinCat
Definition	df-thinc 49777*	Definition of the class of thin categories, or posetal categories, whose hom-sets each contain at most one morphism. Example 3.26(2) of [Adamek] p. 33. "ThinCat" was taken instead of "PosCat" because the latter might mean the category of posets. (Contributed by Zhi Wang, 17-Sep-2024.)
⊢ ThinCat = {𝑐 ∈ Cat ∣ [(Base‘𝑐) / 𝑏][(Hom ‘𝑐) / ℎ]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∃*𝑓 𝑓 ∈ (𝑥ℎ𝑦)}
Theorem	isthinc 49778*	The predicate "is a thin category". (Contributed by Zhi Wang, 17-Sep-2024.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    ⇒   ⊢ (𝐶 ∈ ThinCat ↔ (𝐶 ∈ Cat ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦)))
Theorem	isthinc2 49779*	A thin category is a category in which all hom-sets have cardinality less than or equal to the cardinality of 1o. (Contributed by Zhi Wang, 17-Sep-2024.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    ⇒   ⊢ (𝐶 ∈ ThinCat ↔ (𝐶 ∈ Cat ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐻𝑦) ≼ 1o))
Theorem	isthinc3 49780*	A thin category is a category in which, given a pair of objects 𝑥 and 𝑦 and any two morphisms 𝑓, 𝑔 from 𝑥 to 𝑦, the morphisms are equal. (Contributed by Zhi Wang, 17-Sep-2024.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    ⇒   ⊢ (𝐶 ∈ ThinCat ↔ (𝐶 ∈ Cat ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑥𝐻𝑦)𝑓 = 𝑔))
Theorem	thincc 49781	A thin category is a category. (Contributed by Zhi Wang, 17-Sep-2024.)
⊢ (𝐶 ∈ ThinCat → 𝐶 ∈ Cat)
Theorem	thinccd 49782	A thin category is a category (deduction form). (Contributed by Zhi Wang, 24-Sep-2024.)
⊢ (𝜑 → 𝐶 ∈ ThinCat)    ⇒   ⊢ (𝜑 → 𝐶 ∈ Cat)
Theorem	thincssc 49783	A thin category is a category. (Contributed by Zhi Wang, 17-Sep-2024.)
⊢ ThinCat ⊆ Cat
Theorem	isthincd2lem1 49784*	Lemma for isthincd2 49796 and thincmo2 49785. (Contributed by Zhi Wang, 17-Sep-2024.)
⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))    &   ⊢ (𝜑 → 𝐺 ∈ (𝑋𝐻𝑌))    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦))    ⇒   ⊢ (𝜑 → 𝐹 = 𝐺)
Theorem	thincmo2 49785	Morphisms in the same hom-set are identical. (Contributed by Zhi Wang, 17-Sep-2024.)
⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))    &   ⊢ (𝜑 → 𝐺 ∈ (𝑋𝐻𝑌))    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ ThinCat)    ⇒   ⊢ (𝜑 → 𝐹 = 𝐺)
Theorem	thinchom 49786	A non-empty hom-set of a thin category is given by its element. (Contributed by Zhi Wang, 20-Oct-2025.)
⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ ThinCat)    ⇒   ⊢ (𝜑 → (𝑋𝐻𝑌) = {𝐹})
Theorem	thincmo 49787*	There is at most one morphism in each hom-set. (Contributed by Zhi Wang, 21-Sep-2024.)
⊢ (𝜑 → 𝐶 ∈ ThinCat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    ⇒   ⊢ (𝜑 → ∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌))
Theorem	thincmoALT 49788*	Alternate proof of thincmo 49787. (Contributed by Zhi Wang, 21-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → 𝐶 ∈ ThinCat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    ⇒   ⊢ (𝜑 → ∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌))
Theorem	thincmod 49789*	At most one morphism in each hom-set (deduction form). (Contributed by Zhi Wang, 21-Sep-2024.)
⊢ (𝜑 → 𝐶 ∈ ThinCat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐶))    &   ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶))    ⇒   ⊢ (𝜑 → ∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌))
Theorem	thincn0eu 49790*	In a thin category, a hom-set being non-empty is equivalent to having a unique element. (Contributed by Zhi Wang, 21-Sep-2024.)
⊢ (𝜑 → 𝐶 ∈ ThinCat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐶))    &   ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶))    ⇒   ⊢ (𝜑 → ((𝑋𝐻𝑌) ≠ ∅ ↔ ∃!𝑓 𝑓 ∈ (𝑋𝐻𝑌)))
Theorem	thincid 49791	In a thin category, a morphism from an object to itself is an identity morphism. (Contributed by Zhi Wang, 24-Sep-2024.)
⊢ (𝜑 → 𝐶 ∈ ThinCat)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ 1 = (Id‘𝐶)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑋))    ⇒   ⊢ (𝜑 → 𝐹 = ( 1 ‘𝑋))
Theorem	thincmon 49792	In a thin category, all morphisms are monomorphisms. Example 7.33(9) of [Adamek] p. 110. The converse does not hold. See grptcmon 49952. (Contributed by Zhi Wang, 24-Sep-2024.)
⊢ (𝜑 → 𝐶 ∈ ThinCat)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ 𝑀 = (Mono‘𝐶)    ⇒   ⊢ (𝜑 → (𝑋𝑀𝑌) = (𝑋𝐻𝑌))
Theorem	thincepi 49793	In a thin category, all morphisms are epimorphisms. The converse does not hold. See grptcepi 49953. (Contributed by Zhi Wang, 24-Sep-2024.)
⊢ (𝜑 → 𝐶 ∈ ThinCat)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ 𝐸 = (Epi‘𝐶)    ⇒   ⊢ (𝜑 → (𝑋𝐸𝑌) = (𝑋𝐻𝑌))
Theorem	isthincd2lem2 49794*	Lemma for isthincd2 49796. (Contributed by Zhi Wang, 17-Sep-2024.)
⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))    &   ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑍))    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐻𝑧))    ⇒   ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) ∈ (𝑋𝐻𝑍))
Theorem	isthincd 49795*	The predicate "is a thin category" (deduction form). (Contributed by Zhi Wang, 17-Sep-2024.)
⊢ (𝜑 → 𝐵 = (Base‘𝐶))    &   ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦))    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    ⇒   ⊢ (𝜑 → 𝐶 ∈ ThinCat)
Theorem	isthincd2 49796*	The predicate "𝐶 is a thin category" without knowing 𝐶 is a category (deduction form). The identity arrow operator is also provided as a byproduct. (Contributed by Zhi Wang, 17-Sep-2024.)
⊢ (𝜑 → 𝐵 = (Base‘𝐶))    &   ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦))    &   ⊢ (𝜑 → · = (comp‘𝐶))    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜓 ↔ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧))))    &   ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 1 ∈ (𝑦𝐻𝑦))    &   ⊢ ((𝜑 ∧ 𝜓) → (𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐻𝑧))    ⇒   ⊢ (𝜑 → (𝐶 ∈ ThinCat ∧ (Id‘𝐶) = (𝑦 ∈ 𝐵 ↦ 1 )))
Theorem	oppcthin 49797	The opposite category of a thin category is thin. (Contributed by Zhi Wang, 29-Sep-2024.)
⊢ 𝑂 = (oppCat‘𝐶)    ⇒   ⊢ (𝐶 ∈ ThinCat → 𝑂 ∈ ThinCat)
Theorem	oppcthinco 49798	If the opposite category of a thin category has the same base and hom-sets as the original category, then it has the same composition operation as the original category. (Contributed by Zhi Wang, 16-Oct-2025.)
⊢ 𝑂 = (oppCat‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ ThinCat)    &   ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝑂))    ⇒   ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝑂))
Theorem	oppcthinendc 49799*	The opposite category of a thin category whose morphisms are all endomorphisms has the same base, hom-sets (oppcendc 49377) and composition operation as the original category. (Contributed by Zhi Wang, 16-Oct-2025.)
⊢ 𝑂 = (oppCat‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ ThinCat)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 ≠ 𝑦 → (𝑥𝐻𝑦) = ∅))    ⇒   ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝑂))
Theorem	oppcthinendcALT 49800*	Alternate proof of oppcthinendc 49799. (Contributed by Zhi Wang, 16-Oct-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝑂 = (oppCat‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ ThinCat)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 ≠ 𝑦 → (𝑥𝐻𝑦) = ∅))    ⇒   ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝑂))