P. 499 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 49801-49900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	fucofulem2 49801*	Lemma for proving functor theorems. Maybe consider eufnfv 7173 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 49802	Equivalence of product functor. (Contributed by Zhi Wang, 29-Sep-2025.)
⊢ (⟨⟨𝐾, 𝐿⟩, ⟨𝐹, 𝐺⟩⟩ ∈ (𝑆 × 𝑅) ↔ (𝐾𝑆𝐿 ∧ 𝐹𝑅𝐺))
Theorem	fuco2eld 49803	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 49804	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 49805	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 49806	Extend class notation with functor composition bifunctors.
class ∘F
Definition	df-fuco 49807*	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 49849). The object part maps two functors to their composition (fuco11 49816 and fuco11b 49827). The morphism part defines the "composition" of two natural transformations (fuco22 49829) into another natural transformation (fuco22nat 49836) such that a "cube-like" diagram commutes. The naturality property also gives an alternate definition (fuco23a 49842). Note that such "composition" is different from fucco 17923 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 49808*	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 49809*	Value of the function giving the functor composition bifunctor. Hypotheses fucofval.c and fucofval.d are not redundant (fucofvalne 49815). (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 49810	A functor composition bifunctor is an ordered pair. Enables 1st2ndb 7971. (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 49811	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 49812	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 49813*	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 49814	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 49815*	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 49816	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 49817	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 49818*	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 49819*	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 49820	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 49821	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 49822	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 49823	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 49824	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 49825	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 49826*	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 49827	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 49828	Alternate proof of fuco11b 49827. (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 49829*	The morphism part of the functor composition bifunctor. See also fuco22a 49840. (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 49830	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 49831	The morphism part of the functor composition bifunctor. See also fuco23a 49842. (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 49832	Lemma for fuco22nat 49836. 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 49833	Lemma for fuco22nat 49836. The commutative square of natural transformation 𝐵 in category 𝐸, combined with the commutative square of fuco22natlem1 49832. (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 49834	Combine fuco22natlem2 49833 with fuco23 49831. (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 49835	The composed natural transformation is a natural transformation. Use fuco22nat 49836 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 49836	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 49837	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 49838	Each identity morphism in the source category is mapped to the corresponding identity morphism in the target category. See also fucoid2 49839. (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 49839	Each identity morphism in the source category is mapped to the corresponding identity morphism in the target category. See also fucoid 49838. (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 49840*	The morphism part of the functor composition bifunctor. See also fuco22 49829. (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 49841	The naturality property (nati 17916) 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 49842	The morphism part of the functor composition bifunctor. An alternate definition of ∘F. See also fuco23 49831. (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 49843	Lemma for fucoco 49847. 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 49844*	Lemma for fucoco 49847. 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 49845*	Lemma for fucoco 49847. 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 49846*	Lemma for fucoco 49847. 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 49847	Composition in the source category is mapped to composition in the target. See also fucoco2 49848. (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 49848	Composition in the source category is mapped to composition in the target. See also fucoco 49847. (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 49849	The functor composition bifunctor is a functor. See also fucofunca 49850. 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 49963), for example, (SetCat‘1o) (the trivial category, setc1oterm 49981), 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 49831. 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 49807 up to fuco23 49831, but ((𝐵(𝑈𝑃𝑉)𝐴)‘𝑋) might be mapped to a different morphism in category 𝐸. See fucofulem2 49801 for some insights. 2. fuco22nat 49836, fucoid 49838, and fucoco 49847 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 49850	The functor composition bifunctor is a functor. See also fucofunc 49849. (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 49851*	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 49852*	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 49853	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 49854*	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 49855	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 49856	Lemma for precofval 49857 to enable catlid 17640 or catrid 17641. (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 49857*	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 49858*	Alternate proof of precofval 49857. (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 49859*	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 49860	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 49861*	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 49862*	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 49863	Extend class notation with pre-composition functors.
class −∘F
Definition	df-prcof 49864*	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 49861). But such definition requires an explicit third category. prcoftposcurfuco 49873 and prcoftposcurfucoa 49874 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 49865	The domain of −∘F is a relation. (Contributed by Zhi Wang, 2-Nov-2025.)
⊢ Rel dom −∘F
Theorem	prcofvalg 49866*	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 49867*	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 49868*	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 49869	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 49870	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 49871	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 49872	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 49873	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 49874	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 49875	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 49876	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 49877	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 49878	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 49879*	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 49880*	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 49881	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 49882	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 49883	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 49884	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 49885	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 49886	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 49887	A morphism of the category of categories (in a universe) is a functor. See df-catc 18057 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 18061). (Contributed by Zhi Wang, 14-Nov-2025.)
⊢ 𝐶 = (CatCat‘𝑈)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))    ⇒   ⊢ (𝜑 → 𝐹 ∈ (𝑋 Func 𝑌))
Theorem	catcsect 49888	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 49889	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 49890	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 49891	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 49892	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 49893	The object part of the op functor on functor categories. Lemma for fucoppc 49900. (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 49894	The object part of the op functor on functor categories. Lemma for oppfdiag 49906. (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 49895*	The morphism part of the op functor on functor categories. Lemma for fucoppc 49900. (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 49896*	The morphism part of the op functor on functor categories. Lemma for fucoppc 49900. (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 49897	Lemma for fucoppc 49900. (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 49898*	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 49899*	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 49900*	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)