P. 504 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 50301-50400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	elxpcbasex1ALT 50301	Alternate proof of elxpcbasex1 50300. (Contributed by Zhi Wang, 8-Oct-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝑇 = (𝐶 ×c 𝐷)    &   ⊢ 𝐵 = (Base‘𝑇)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → 𝐶 ∈ V)
Theorem	elxpcbasex2 50302	A non-empty base set of the product category indicates the existence of the second factor of the product category. (Contributed by Zhi Wang, 8-Oct-2025.) (Proof shortened by SN, 15-Oct-2025.)
⊢ 𝑇 = (𝐶 ×c 𝐷)    &   ⊢ 𝐵 = (Base‘𝑇)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → 𝐷 ∈ V)
Theorem	elxpcbasex2ALT 50303	Alternate proof of elxpcbasex2 50302. (Contributed by Zhi Wang, 8-Oct-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝑇 = (𝐶 ×c 𝐷)    &   ⊢ 𝐵 = (Base‘𝑇)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → 𝐷 ∈ V)
Theorem	xpcfucbas 50304	The base set of the product of two categories of functors. (Contributed by Zhi Wang, 1-Oct-2025.)
⊢ 𝑇 = ((𝐵 FuncCat 𝐶) ×c (𝐷 FuncCat 𝐸))    ⇒   ⊢ ((𝐵 Func 𝐶) × (𝐷 Func 𝐸)) = (Base‘𝑇)
Theorem	xpcfuchomfval 50305*	Set of morphisms of the binary product of categories of functors. (Contributed by Zhi Wang, 1-Oct-2025.)
⊢ 𝑇 = ((𝐵 FuncCat 𝐶) ×c (𝐷 FuncCat 𝐸))    &   ⊢ 𝐴 = (Base‘𝑇)    &   ⊢ 𝐾 = (Hom ‘𝑇)    ⇒   ⊢ 𝐾 = (𝑢 ∈ 𝐴, 𝑣 ∈ 𝐴 ↦ (((1st ‘𝑢)(𝐵 Nat 𝐶)(1st ‘𝑣)) × ((2nd ‘𝑢)(𝐷 Nat 𝐸)(2nd ‘𝑣))))
Theorem	xpcfuchom 50306	Set of morphisms of the binary product of categories of functors. (Contributed by Zhi Wang, 1-Oct-2025.)
⊢ 𝑇 = ((𝐵 FuncCat 𝐶) ×c (𝐷 FuncCat 𝐸))    &   ⊢ 𝐴 = (Base‘𝑇)    &   ⊢ 𝐾 = (Hom ‘𝑇)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐴)    ⇒   ⊢ (𝜑 → (𝑋𝐾𝑌) = (((1st ‘𝑋)(𝐵 Nat 𝐶)(1st ‘𝑌)) × ((2nd ‘𝑋)(𝐷 Nat 𝐸)(2nd ‘𝑌))))
Theorem	xpcfuchom2 50307	Value of the set of morphisms in the binary product of categories of functors. (Contributed by Zhi Wang, 1-Oct-2025.)
⊢ 𝑇 = ((𝐵 FuncCat 𝐶) ×c (𝐷 FuncCat 𝐸))    &   ⊢ (𝜑 → 𝑀 ∈ (𝐵 Func 𝐶))    &   ⊢ (𝜑 → 𝑁 ∈ (𝐷 Func 𝐸))    &   ⊢ (𝜑 → 𝑃 ∈ (𝐵 Func 𝐶))    &   ⊢ (𝜑 → 𝑄 ∈ (𝐷 Func 𝐸))    &   ⊢ 𝐾 = (Hom ‘𝑇)    ⇒   ⊢ (𝜑 → (⟨𝑀, 𝑁⟩𝐾⟨𝑃, 𝑄⟩) = ((𝑀(𝐵 Nat 𝐶)𝑃) × (𝑁(𝐷 Nat 𝐸)𝑄)))
Theorem	xpcfucco2 50308	Value of composition in the binary product of categories of functors. (Contributed by Zhi Wang, 1-Oct-2025.)
⊢ 𝑇 = ((𝐵 FuncCat 𝐶) ×c (𝐷 FuncCat 𝐸))    &   ⊢ 𝑂 = (comp‘𝑇)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑀(𝐵 Nat 𝐶)𝑃))    &   ⊢ (𝜑 → 𝐺 ∈ (𝑁(𝐷 Nat 𝐸)𝑄))    &   ⊢ (𝜑 → 𝐾 ∈ (𝑃(𝐵 Nat 𝐶)𝑅))    &   ⊢ (𝜑 → 𝐿 ∈ (𝑄(𝐷 Nat 𝐸)𝑆))    ⇒   ⊢ (𝜑 → (⟨𝐾, 𝐿⟩(⟨⟨𝑀, 𝑁⟩, ⟨𝑃, 𝑄⟩⟩𝑂⟨𝑅, 𝑆⟩)⟨𝐹, 𝐺⟩) = ⟨(𝐾(⟨𝑀, 𝑃⟩(comp‘(𝐵 FuncCat 𝐶))𝑅)𝐹), (𝐿(⟨𝑁, 𝑄⟩(comp‘(𝐷 FuncCat 𝐸))𝑆)𝐺)⟩)
Theorem	xpcfuccocl 50309	The composition of two natural transformations is a natural transformation. (Contributed by Zhi Wang, 1-Oct-2025.)
⊢ 𝑇 = ((𝐵 FuncCat 𝐶) ×c (𝐷 FuncCat 𝐸))    &   ⊢ 𝑂 = (comp‘𝑇)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑀(𝐵 Nat 𝐶)𝑃))    &   ⊢ (𝜑 → 𝐺 ∈ (𝑁(𝐷 Nat 𝐸)𝑄))    &   ⊢ (𝜑 → 𝐾 ∈ (𝑃(𝐵 Nat 𝐶)𝑅))    &   ⊢ (𝜑 → 𝐿 ∈ (𝑄(𝐷 Nat 𝐸)𝑆))    ⇒   ⊢ (𝜑 → (⟨𝐾, 𝐿⟩(⟨⟨𝑀, 𝑁⟩, ⟨𝑃, 𝑄⟩⟩𝑂⟨𝑅, 𝑆⟩)⟨𝐹, 𝐺⟩) ∈ ((𝑀(𝐵 Nat 𝐶)𝑅) × (𝑁(𝐷 Nat 𝐸)𝑆)))
Theorem	xpcfucco3 50310*	Value of composition in the binary product of categories of functors; expressed explicitly. (Contributed by Zhi Wang, 1-Oct-2025.)
⊢ 𝑇 = ((𝐵 FuncCat 𝐶) ×c (𝐷 FuncCat 𝐸))    &   ⊢ 𝑂 = (comp‘𝑇)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑀(𝐵 Nat 𝐶)𝑃))    &   ⊢ (𝜑 → 𝐺 ∈ (𝑁(𝐷 Nat 𝐸)𝑄))    &   ⊢ (𝜑 → 𝐾 ∈ (𝑃(𝐵 Nat 𝐶)𝑅))    &   ⊢ (𝜑 → 𝐿 ∈ (𝑄(𝐷 Nat 𝐸)𝑆))    &   ⊢ 𝑋 = (Base‘𝐵)    &   ⊢ 𝑌 = (Base‘𝐷)    &   ⊢ · = (comp‘𝐶)    &   ⊢ ∙ = (comp‘𝐸)    ⇒   ⊢ (𝜑 → (⟨𝐾, 𝐿⟩(⟨⟨𝑀, 𝑁⟩, ⟨𝑃, 𝑄⟩⟩𝑂⟨𝑅, 𝑆⟩)⟨𝐹, 𝐺⟩) = ⟨(𝑥 ∈ 𝑋 ↦ ((𝐾‘𝑥)(⟨((1st ‘𝑀)‘𝑥), ((1st ‘𝑃)‘𝑥)⟩ · ((1st ‘𝑅)‘𝑥))(𝐹‘𝑥))), (𝑦 ∈ 𝑌 ↦ ((𝐿‘𝑦)(⟨((1st ‘𝑁)‘𝑦), ((1st ‘𝑄)‘𝑦)⟩ ∙ ((1st ‘𝑆)‘𝑦))(𝐺‘𝑦)))⟩)
21.51.15.14  Swap functors
Syntax	cswapf 50311	Extend class notation with the class of swap functors.
class swapF
Definition	df-swapf 50312*	Define the swap functor from (𝐶 ×c 𝐷) to (𝐷 ×c 𝐶) by swapping all objects (swapf1 50324) and morphisms (swapf2 50326) . Such functor is called a "swap functor" in https://arxiv.org/pdf/2302.07810 50326 or a "twist functor" in https://arxiv.org/pdf/2508.01886 50326, the latter of which finds its counterpart as "twisting map" in https://arxiv.org/pdf/2411.04102 50326 for tensor product of algebras. The "swap functor" or "twisting map" is often denoted as a small tau 𝜏 in literature. However, the term "twist functor" is defined differently in https://arxiv.org/pdf/1208.4046 50326 and thus not adopted here. tpos I depends on more mathbox theorems, and thus are not adopted here. See dfswapf2 50313 for an alternate definition. (Contributed by Zhi Wang, 7-Oct-2025.)
⊢ swapF = (𝑐 ∈ V, 𝑑 ∈ V ↦ ⦋(𝑐 ×c 𝑑) / 𝑠⦌⦋(Base‘𝑠) / 𝑏⦌⦋(Hom ‘𝑠) / ℎ⦌⟨(𝑥 ∈ 𝑏 ↦ ∪ ◡{𝑥}), (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (𝑓 ∈ (𝑢ℎ𝑣) ↦ ∪ ◡{𝑓}))⟩)
Theorem	dfswapf2 50313*	Alternate definition of swapF (df-swapf 50312). (Contributed by Zhi Wang, 9-Oct-2025.)
⊢ swapF = (𝑐 ∈ V, 𝑑 ∈ V ↦ ⦋(𝑐 ×c 𝑑) / 𝑠⦌⦋(Base‘𝑠) / 𝑏⦌⦋(Hom ‘𝑠) / ℎ⦌⟨(tpos I ↾ 𝑏), (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (tpos I ↾ (𝑢ℎ𝑣)))⟩)
Theorem	swapfval 50314*	Value of the swap functor. (Contributed by Zhi Wang, 7-Oct-2025.)
⊢ (𝜑 → 𝐶 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑉)    &   ⊢ 𝑆 = (𝐶 ×c 𝐷)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝐻 = (Hom ‘𝑆))    ⇒   ⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨(𝑥 ∈ 𝐵 ↦ ∪ ◡{𝑥}), (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑓 ∈ (𝑢𝐻𝑣) ↦ ∪ ◡{𝑓}))⟩)
Theorem	swapfelvv 50315	A swap functor is an ordered pair. (Contributed by Zhi Wang, 7-Oct-2025.)
⊢ (𝜑 → 𝐶 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝐶 swapF 𝐷) ∈ (V × V))
Theorem	swapf2fvala 50316*	The morphism part of the swap functor. See also swapf2fval 50317. (Contributed by Zhi Wang, 7-Oct-2025.)
⊢ (𝜑 → 𝐶 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑉)    &   ⊢ 𝑆 = (𝐶 ×c 𝐷)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝐻 = (Hom ‘𝑆))    ⇒   ⊢ (𝜑 → (2nd ‘(𝐶 swapF 𝐷)) = (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑓 ∈ (𝑢𝐻𝑣) ↦ ∪ ◡{𝑓})))
Theorem	swapf2fval 50317*	The morphism part of the swap functor. See also swapf2fvala 50316. (Contributed by Zhi Wang, 7-Oct-2025.)
⊢ (𝜑 → 𝐶 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑉)    &   ⊢ 𝑆 = (𝐶 ×c 𝐷)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝐻 = (Hom ‘𝑆))    &   ⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨𝑂, 𝑃⟩)    ⇒   ⊢ (𝜑 → 𝑃 = (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑓 ∈ (𝑢𝐻𝑣) ↦ ∪ ◡{𝑓})))
Theorem	swapf1vala 50318*	The object part of the swap functor. See also swapf1val 50319. (Contributed by Zhi Wang, 7-Oct-2025.)
⊢ (𝜑 → 𝐶 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑉)    &   ⊢ 𝑆 = (𝐶 ×c 𝐷)    &   ⊢ 𝐵 = (Base‘𝑆)    ⇒   ⊢ (𝜑 → (1st ‘(𝐶 swapF 𝐷)) = (𝑥 ∈ 𝐵 ↦ ∪ ◡{𝑥}))
Theorem	swapf1val 50319*	The object part of the swap functor. See also swapf1vala 50318. (Contributed by Zhi Wang, 7-Oct-2025.)
⊢ (𝜑 → 𝐶 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑉)    &   ⊢ 𝑆 = (𝐶 ×c 𝐷)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨𝑂, 𝑃⟩)    ⇒   ⊢ (𝜑 → 𝑂 = (𝑥 ∈ 𝐵 ↦ ∪ ◡{𝑥}))
Theorem	swapf2fn 50320	The morphism part of the swap functor is a function on the Cartesian square of the base set. (Contributed by Zhi Wang, 7-Oct-2025.)
⊢ (𝜑 → 𝐶 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑉)    &   ⊢ 𝑆 = (𝐶 ×c 𝐷)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨𝑂, 𝑃⟩)    ⇒   ⊢ (𝜑 → 𝑃 Fn (𝐵 × 𝐵))
Theorem	swapf1a 50321	The object part of the swap functor swaps the objects. (Contributed by Zhi Wang, 7-Oct-2025.)
⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨𝑂, 𝑃⟩)    &   ⊢ 𝑆 = (𝐶 ×c 𝐷)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑂‘𝑋) = ⟨(2nd ‘𝑋), (1st ‘𝑋)⟩)
Theorem	swapf2vala 50322*	The morphism part of the swap functor swaps the morphisms. (Contributed by Zhi Wang, 7-Oct-2025.)
⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨𝑂, 𝑃⟩)    &   ⊢ 𝑆 = (𝐶 ×c 𝐷)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐻 = (Hom ‘𝑆))    ⇒   ⊢ (𝜑 → (𝑋𝑃𝑌) = (𝑓 ∈ (𝑋𝐻𝑌) ↦ ∪ ◡{𝑓}))
Theorem	swapf2a 50323	The morphism part of the swap functor swaps the morphisms. (Contributed by Zhi Wang, 7-Oct-2025.)
⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨𝑂, 𝑃⟩)    &   ⊢ 𝑆 = (𝐶 ×c 𝐷)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐻 = (Hom ‘𝑆))    &   ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))    ⇒   ⊢ (𝜑 → ((𝑋𝑃𝑌)‘𝐹) = ⟨(2nd ‘𝐹), (1st ‘𝐹)⟩)
Theorem	swapf1 50324	The object part of the swap functor swaps the objects. (Contributed by Zhi Wang, 7-Oct-2025.)
⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨𝑂, 𝑃⟩)    &   ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))    &   ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐷))    ⇒   ⊢ (𝜑 → (𝑋𝑂𝑌) = ⟨𝑌, 𝑋⟩)
Theorem	swapf2val 50325*	The morphism part of the swap functor swaps the morphisms. (Contributed by Zhi Wang, 7-Oct-2025.)
⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨𝑂, 𝑃⟩)    &   ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))    &   ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐷))    &   ⊢ (𝜑 → 𝑍 ∈ (Base‘𝐶))    &   ⊢ (𝜑 → 𝑊 ∈ (Base‘𝐷))    &   ⊢ 𝑆 = (𝐶 ×c 𝐷)    &   ⊢ (𝜑 → 𝐻 = (Hom ‘𝑆))    ⇒   ⊢ (𝜑 → (⟨𝑋, 𝑌⟩𝑃⟨𝑍, 𝑊⟩) = (𝑓 ∈ (⟨𝑋, 𝑌⟩𝐻⟨𝑍, 𝑊⟩) ↦ ∪ ◡{𝑓}))
Theorem	swapf2 50326	The morphism part of the swap functor swaps the morphisms. (Contributed by Zhi Wang, 7-Oct-2025.)
⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨𝑂, 𝑃⟩)    &   ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))    &   ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐷))    &   ⊢ (𝜑 → 𝑍 ∈ (Base‘𝐶))    &   ⊢ (𝜑 → 𝑊 ∈ (Base‘𝐷))    &   ⊢ (𝜑 → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑍))    &   ⊢ (𝜑 → 𝐺 ∈ (𝑌(Hom ‘𝐷)𝑊))    ⇒   ⊢ (𝜑 → (𝐹(⟨𝑋, 𝑌⟩𝑃⟨𝑍, 𝑊⟩)𝐺) = ⟨𝐺, 𝐹⟩)
Theorem	swapf1f1o 50327	The object part of the swap functor is a bijection between base sets. (Contributed by Zhi Wang, 8-Oct-2025.)
⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨𝑂, 𝑃⟩)    &   ⊢ 𝑆 = (𝐶 ×c 𝐷)    &   ⊢ 𝑇 = (𝐷 ×c 𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑉)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ 𝐴 = (Base‘𝑇)    ⇒   ⊢ (𝜑 → 𝑂:𝐵–1-1-onto→𝐴)
Theorem	swapf2f1o 50328	The morphism part of the swap functor is a bijection between hom-sets. (Contributed by Zhi Wang, 8-Oct-2025.)
⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨𝑂, 𝑃⟩)    &   ⊢ 𝑆 = (𝐶 ×c 𝐷)    &   ⊢ 𝑇 = (𝐷 ×c 𝐶)    &   ⊢ 𝐻 = (Hom ‘𝑆)    &   ⊢ 𝐽 = (Hom ‘𝑇)    &   ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))    &   ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐷))    &   ⊢ (𝜑 → 𝑍 ∈ (Base‘𝐶))    &   ⊢ (𝜑 → 𝑊 ∈ (Base‘𝐷))    ⇒   ⊢ (𝜑 → (⟨𝑋, 𝑌⟩𝑃⟨𝑍, 𝑊⟩):(⟨𝑋, 𝑌⟩𝐻⟨𝑍, 𝑊⟩)–1-1-onto→(⟨𝑌, 𝑋⟩𝐽⟨𝑊, 𝑍⟩))
Theorem	swapf2f1oa 50329	The morphism part of the swap functor is a bijection between hom-sets. (Contributed by Zhi Wang, 9-Oct-2025.)
⊢ (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	swapf2f1oaALT 50330	Alternate proof of swapf2f1oa 50329. (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 50331	Each identity morphism in the source category is mapped to the corresponding identity morphism in the target category. See also swapfida 50332. (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 50332	Each identity morphism in the source category is mapped to the corresponding identity morphism in the target category. See also swapfid 50331. (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 50333	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 50334	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 50335	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 50336	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 50337	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 50338	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)
21.51.15.15  Functor evaluation
Theorem	oppc1stflem 50339*	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 50340	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 50341	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 50342	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 50343	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 50344	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))
21.51.15.16  Transposed curry functors
Theorem	cofuswapfcl 50345	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 50346	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 50347	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 50348	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 50349	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 50350	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 50351*	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 50352*	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 50353	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 50354	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 50355	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))
21.51.15.17  Constant functors
Theorem	diag1 50356*	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 50357*	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 50358	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 50359 and f1fveq 7258. (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 50359	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 50360	Lemma for diag2f1 50361. The converse is trivial (fveq2 6877). (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 50361	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)))
21.51.15.18  Functor composition bifunctors
Theorem	fucofulem1 50362	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 50363*	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 50364	Equivalence of product functor. (Contributed by Zhi Wang, 29-Sep-2025.)
⊢ (⟨⟨𝐾, 𝐿⟩, ⟨𝐹, 𝐺⟩⟩ ∈ (𝑆 × 𝑅) ↔ (𝐾𝑆𝐿 ∧ 𝐹𝑅𝐺))
Theorem	fuco2eld 50365	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 50366	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 50367	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 50368	Extend class notation with functor composition bifunctors.
class ∘F
Definition	df-fuco 50369*	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 50411). The object part maps two functors to their composition (fuco11 50378 and fuco11b 50389). The morphism part defines the "composition" of two natural transformations (fuco22 50391) into another natural transformation (fuco22nat 50398) such that a "cube-like" diagram commutes. The naturality property also gives an alternate definition (fuco23a 50404). Note that such "composition" is different from fucco 18120 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 50370*	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 50371*	Value of the function giving the functor composition bifunctor. Hypotheses fucofval.c and fucofval.d are not redundant (fucofvalne 50377). (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 50372	A functor composition bifunctor is an ordered pair. Enables 1st2ndb 8030. (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 50373	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 50374	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 50375*	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 50376	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 50377*	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 50378	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 50379	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 50380*	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 50381*	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 50382	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 50383	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 50384	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 50385	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 50386	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 50387	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 50388*	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 50389	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 50390	Alternate proof of fuco11b 50389. (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 50391*	The morphism part of the functor composition bifunctor. See also fuco22a 50402. (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 50392	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 50393	The morphism part of the functor composition bifunctor. See also fuco23a 50404. (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 50394	Lemma for fuco22nat 50398. 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 50395	Lemma for fuco22nat 50398. The commutative square of natural transformation 𝐵 in category 𝐸, combined with the commutative square of fuco22natlem1 50394. (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 50396	Combine fuco22natlem2 50395 with fuco23 50393. (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 50397	The composed natural transformation is a natural transformation. Use fuco22nat 50398 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 50398	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 50399	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 50400	Each identity morphism in the source category is mapped to the corresponding identity morphism in the target category. See also fucoid2 50401. (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)))