P. 180 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 17901-18000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	fullresc 17901	The category formed by structure restriction is the same as the category restriction. (Contributed by Mario Carneiro, 5-Jan-2017.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Homf ‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝐵)    &   ⊢ 𝐷 = (𝐶 ↾s 𝑆)    &   ⊢ 𝐸 = (𝐶 ↾cat (𝐻 ↾ (𝑆 × 𝑆)))    ⇒   ⊢ (𝜑 → ((Homf ‘𝐷) = (Homf ‘𝐸) ∧ (compf‘𝐷) = (compf‘𝐸)))
Theorem	resscat 17902	A category restricted to a smaller set of objects is a category. (Contributed by Mario Carneiro, 6-Jan-2017.)
⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → (𝐶 ↾s 𝑆) ∈ Cat)
Theorem	subsubc 17903	A subcategory of a subcategory is a subcategory. (Contributed by Mario Carneiro, 6-Jan-2017.)
⊢ 𝐷 = (𝐶 ↾cat 𝐻)    ⇒   ⊢ (𝐻 ∈ (Subcat‘𝐶) → (𝐽 ∈ (Subcat‘𝐷) ↔ (𝐽 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻)))
8.1.7  Functors
Syntax	cfunc 17904	Extend class notation with the class of all functors.
class Func
Syntax	cidfu 17905	Extend class notation with identity functor.
class idfunc
Syntax	ccofu 17906	Extend class notation with functor composition.
class ∘func
Syntax	cresf 17907	Extend class notation to include restriction of a functor to a subcategory.
class ↾f
Definition	df-func 17908*	Function returning all the functors from a category 𝑡 to a category 𝑢. Definition 3.17 of [Adamek] p. 29, and definition in [Lang] p. 62 ("covariant functor"). Intuitively a functor associates any morphism of 𝑡 to a morphism of 𝑢, any object of 𝑡 to an object of 𝑢, and respects the identity, the composition, the domain and the codomain. Here to capture the idea that a functor associates any object of 𝑡 to an object of 𝑢 we write it associates any identity of 𝑡 to an identity of 𝑢 which simplifies the definition. According to remark 3.19 in [Adamek] p. 30, "a functor F : A -> B is technically a family of functions; one from Ob(A) to Ob(B) [here: f, called "the object part" in the following], and for each pair (A,A') of A-objects, one from hom(A,A') to hom(FA, FA') [here: g, called "the morphism part" in the following]". (Contributed by FL, 10-Feb-2008.) (Revised by Mario Carneiro, 2-Jan-2017.)
⊢ Func = (𝑡 ∈ Cat, 𝑢 ∈ Cat ↦ {⟨𝑓, 𝑔⟩ ∣ [(Base‘𝑡) / 𝑏](𝑓:𝑏⟶(Base‘𝑢) ∧ 𝑔 ∈ X𝑧 ∈ (𝑏 × 𝑏)(((𝑓‘(1st ‘𝑧))(Hom ‘𝑢)(𝑓‘(2nd ‘𝑧))) ↑m ((Hom ‘𝑡)‘𝑧)) ∧ ∀𝑥 ∈ 𝑏 (((𝑥𝑔𝑥)‘((Id‘𝑡)‘𝑥)) = ((Id‘𝑢)‘(𝑓‘𝑥)) ∧ ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ∀𝑚 ∈ (𝑥(Hom ‘𝑡)𝑦)∀𝑛 ∈ (𝑦(Hom ‘𝑡)𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦⟩(comp‘𝑡)𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓‘𝑥), (𝑓‘𝑦)⟩(comp‘𝑢)(𝑓‘𝑧))((𝑥𝑔𝑦)‘𝑚))))})
Definition	df-idfu 17909*	Define the identity functor. (Contributed by Mario Carneiro, 3-Jan-2017.)
⊢ idfunc = (𝑡 ∈ Cat ↦ ⦋(Base‘𝑡) / 𝑏⦌⟨( I ↾ 𝑏), (𝑧 ∈ (𝑏 × 𝑏) ↦ ( I ↾ ((Hom ‘𝑡)‘𝑧)))⟩)
Definition	df-cofu 17910*	Define the composition of two functors. (Contributed by Mario Carneiro, 3-Jan-2017.)
⊢ ∘func = (𝑔 ∈ V, 𝑓 ∈ V ↦ ⟨((1st ‘𝑔) ∘ (1st ‘𝑓)), (𝑥 ∈ dom dom (2nd ‘𝑓), 𝑦 ∈ dom dom (2nd ‘𝑓) ↦ ((((1st ‘𝑓)‘𝑥)(2nd ‘𝑔)((1st ‘𝑓)‘𝑦)) ∘ (𝑥(2nd ‘𝑓)𝑦)))⟩)
Definition	df-resf 17911*	Define the restriction of a functor to a subcategory (analogue of df-res 5700). (Contributed by Mario Carneiro, 6-Jan-2017.)
⊢ ↾f = (𝑓 ∈ V, ℎ ∈ V ↦ ⟨((1st ‘𝑓) ↾ dom dom ℎ), (𝑥 ∈ dom ℎ ↦ (((2nd ‘𝑓)‘𝑥) ↾ (ℎ‘𝑥)))⟩)
Theorem	relfunc 17912	The set of functors is a relation. (Contributed by Mario Carneiro, 2-Jan-2017.)
⊢ Rel (𝐷 Func 𝐸)
Theorem	funcrcl 17913	Reverse closure for a functor. (Contributed by Mario Carneiro, 6-Jan-2017.)
⊢ (𝐹 ∈ (𝐷 Func 𝐸) → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))
Theorem	isfunc 17914*	Value of the set of functors between two categories. (Contributed by Mario Carneiro, 2-Jan-2017.)
⊢ 𝐵 = (Base‘𝐷)    &   ⊢ 𝐶 = (Base‘𝐸)    &   ⊢ 𝐻 = (Hom ‘𝐷)    &   ⊢ 𝐽 = (Hom ‘𝐸)    &   ⊢ 1 = (Id‘𝐷)    &   ⊢ 𝐼 = (Id‘𝐸)    &   ⊢ · = (comp‘𝐷)    &   ⊢ 𝑂 = (comp‘𝐸)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ (𝜑 → 𝐸 ∈ Cat)    ⇒   ⊢ (𝜑 → (𝐹(𝐷 Func 𝐸)𝐺 ↔ (𝐹:𝐵⟶𝐶 ∧ 𝐺 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st ‘𝑧))𝐽(𝐹‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧)) ∧ ∀𝑥 ∈ 𝐵 (((𝑥𝐺𝑥)‘( 1 ‘𝑥)) = (𝐼‘(𝐹‘𝑥)) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩𝑂(𝐹‘𝑧))((𝑥𝐺𝑦)‘𝑚))))))
Theorem	isfuncd 17915*	Deduce that an operation is a functor of categories. (Contributed by Mario Carneiro, 4-Jan-2017.)
⊢ 𝐵 = (Base‘𝐷)    &   ⊢ 𝐶 = (Base‘𝐸)    &   ⊢ 𝐻 = (Hom ‘𝐷)    &   ⊢ 𝐽 = (Hom ‘𝐸)    &   ⊢ 1 = (Id‘𝐷)    &   ⊢ 𝐼 = (Id‘𝐸)    &   ⊢ · = (comp‘𝐷)    &   ⊢ 𝑂 = (comp‘𝐸)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ (𝜑 → 𝐸 ∈ Cat)    &   ⊢ (𝜑 → 𝐹:𝐵⟶𝐶)    &   ⊢ (𝜑 → 𝐺 Fn (𝐵 × 𝐵))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹‘𝑥)𝐽(𝐹‘𝑦)))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((𝑥𝐺𝑥)‘( 1 ‘𝑥)) = (𝐼‘(𝐹‘𝑥)))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑚 ∈ (𝑥𝐻𝑦) ∧ 𝑛 ∈ (𝑦𝐻𝑧))) → ((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩𝑂(𝐹‘𝑧))((𝑥𝐺𝑦)‘𝑚)))    ⇒   ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)
Theorem	funcf1 17916	The object part of a functor is a function on objects. (Contributed by Mario Carneiro, 2-Jan-2017.)
⊢ 𝐵 = (Base‘𝐷)    &   ⊢ 𝐶 = (Base‘𝐸)    &   ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)    ⇒   ⊢ (𝜑 → 𝐹:𝐵⟶𝐶)
Theorem	funcixp 17917*	The morphism part of a functor is a function on homsets. (Contributed by Mario Carneiro, 2-Jan-2017.)
⊢ 𝐵 = (Base‘𝐷)    &   ⊢ 𝐻 = (Hom ‘𝐷)    &   ⊢ 𝐽 = (Hom ‘𝐸)    &   ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)    ⇒   ⊢ (𝜑 → 𝐺 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st ‘𝑧))𝐽(𝐹‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧)))
Theorem	funcf2 17918	The morphism part of a functor is a function on homsets. (Contributed by Mario Carneiro, 2-Jan-2017.)
⊢ 𝐵 = (Base‘𝐷)    &   ⊢ 𝐻 = (Hom ‘𝐷)    &   ⊢ 𝐽 = (Hom ‘𝐸)    &   ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑋𝐺𝑌):(𝑋𝐻𝑌)⟶((𝐹‘𝑋)𝐽(𝐹‘𝑌)))
Theorem	funcfn2 17919	The morphism part of a functor is a function. (Contributed by Mario Carneiro, 3-Jan-2017.)
⊢ 𝐵 = (Base‘𝐷)    &   ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)    ⇒   ⊢ (𝜑 → 𝐺 Fn (𝐵 × 𝐵))
Theorem	funcid 17920	A functor maps each identity to the corresponding identity in the target category. (Contributed by Mario Carneiro, 2-Jan-2017.)
⊢ 𝐵 = (Base‘𝐷)    &   ⊢ 1 = (Id‘𝐷)    &   ⊢ 𝐼 = (Id‘𝐸)    &   ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ((𝑋𝐺𝑋)‘( 1 ‘𝑋)) = (𝐼‘(𝐹‘𝑋)))
Theorem	funcco 17921	A functor maps composition in the source category to composition in the target. (Contributed by Mario Carneiro, 2-Jan-2017.)
⊢ 𝐵 = (Base‘𝐷)    &   ⊢ 𝐻 = (Hom ‘𝐷)    &   ⊢ · = (comp‘𝐷)    &   ⊢ 𝑂 = (comp‘𝐸)    &   ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑀 ∈ (𝑋𝐻𝑌))    &   ⊢ (𝜑 → 𝑁 ∈ (𝑌𝐻𝑍))    ⇒   ⊢ (𝜑 → ((𝑋𝐺𝑍)‘(𝑁(⟨𝑋, 𝑌⟩ · 𝑍)𝑀)) = (((𝑌𝐺𝑍)‘𝑁)(⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩𝑂(𝐹‘𝑍))((𝑋𝐺𝑌)‘𝑀)))
Theorem	funcsect 17922	The image of a section under a functor is a section. (Contributed by Mario Carneiro, 2-Jan-2017.)
⊢ 𝐵 = (Base‘𝐷)    &   ⊢ 𝑆 = (Sect‘𝐷)    &   ⊢ 𝑇 = (Sect‘𝐸)    &   ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑀(𝑋𝑆𝑌)𝑁)    ⇒   ⊢ (𝜑 → ((𝑋𝐺𝑌)‘𝑀)((𝐹‘𝑋)𝑇(𝐹‘𝑌))((𝑌𝐺𝑋)‘𝑁))
Theorem	funcinv 17923	The image of an inverse under a functor is an inverse. (Contributed by Mario Carneiro, 3-Jan-2017.)
⊢ 𝐵 = (Base‘𝐷)    &   ⊢ 𝐼 = (Inv‘𝐷)    &   ⊢ 𝐽 = (Inv‘𝐸)    &   ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑀(𝑋𝐼𝑌)𝑁)    ⇒   ⊢ (𝜑 → ((𝑋𝐺𝑌)‘𝑀)((𝐹‘𝑋)𝐽(𝐹‘𝑌))((𝑌𝐺𝑋)‘𝑁))
Theorem	funciso 17924	The image of an isomorphism under a functor is an isomorphism. Proposition 3.21 of [Adamek] p. 32. (Contributed by Mario Carneiro, 3-Jan-2017.)
⊢ 𝐵 = (Base‘𝐷)    &   ⊢ 𝐼 = (Iso‘𝐷)    &   ⊢ 𝐽 = (Iso‘𝐸)    &   ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑀 ∈ (𝑋𝐼𝑌))    ⇒   ⊢ (𝜑 → ((𝑋𝐺𝑌)‘𝑀) ∈ ((𝐹‘𝑋)𝐽(𝐹‘𝑌)))
Theorem	funcoppc 17925	A functor on categories yields a functor on the opposite categories (in the same direction), see definition 3.41 of [Adamek] p. 39. (Contributed by Mario Carneiro, 4-Jan-2017.)
⊢ 𝑂 = (oppCat‘𝐶)    &   ⊢ 𝑃 = (oppCat‘𝐷)    &   ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)    ⇒   ⊢ (𝜑 → 𝐹(𝑂 Func 𝑃)tpos 𝐺)
Theorem	idfuval 17926*	Value of the identity functor. (Contributed by Mario Carneiro, 3-Jan-2017.)
⊢ 𝐼 = (idfunc‘𝐶)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ 𝐻 = (Hom ‘𝐶)    ⇒   ⊢ (𝜑 → 𝐼 = ⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧)))⟩)
Theorem	idfu2nd 17927	Value of the morphism part of the identity functor. (Contributed by Mario Carneiro, 3-Jan-2017.)
⊢ 𝐼 = (idfunc‘𝐶)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑋(2nd ‘𝐼)𝑌) = ( I ↾ (𝑋𝐻𝑌)))
Theorem	idfu2 17928	Value of the morphism part of the identity functor. (Contributed by Mario Carneiro, 28-Jan-2017.)
⊢ 𝐼 = (idfunc‘𝐶)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))    ⇒   ⊢ (𝜑 → ((𝑋(2nd ‘𝐼)𝑌)‘𝐹) = 𝐹)
Theorem	idfu1st 17929	Value of the object part of the identity functor. (Contributed by Mario Carneiro, 3-Jan-2017.)
⊢ 𝐼 = (idfunc‘𝐶)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    ⇒   ⊢ (𝜑 → (1st ‘𝐼) = ( I ↾ 𝐵))
Theorem	idfu1 17930	Value of the object part of the identity functor. (Contributed by Mario Carneiro, 3-Jan-2017.)
⊢ 𝐼 = (idfunc‘𝐶)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ((1st ‘𝐼)‘𝑋) = 𝑋)
Theorem	idfucl 17931	The identity functor is a functor. Example 3.20(1) of [Adamek] p. 30. (Contributed by Mario Carneiro, 3-Jan-2017.)
⊢ 𝐼 = (idfunc‘𝐶)    ⇒   ⊢ (𝐶 ∈ Cat → 𝐼 ∈ (𝐶 Func 𝐶))
Theorem	cofuval 17932*	Value of the composition of two functors. (Contributed by Mario Carneiro, 3-Jan-2017.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    &   ⊢ (𝜑 → 𝐺 ∈ (𝐷 Func 𝐸))    ⇒   ⊢ (𝜑 → (𝐺 ∘func 𝐹) = ⟨((1st ‘𝐺) ∘ (1st ‘𝐹)), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑦)) ∘ (𝑥(2nd ‘𝐹)𝑦)))⟩)
Theorem	cofu1st 17933	Value of the object part of the functor composition. (Contributed by Mario Carneiro, 3-Jan-2017.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    &   ⊢ (𝜑 → 𝐺 ∈ (𝐷 Func 𝐸))    ⇒   ⊢ (𝜑 → (1st ‘(𝐺 ∘func 𝐹)) = ((1st ‘𝐺) ∘ (1st ‘𝐹)))
Theorem	cofu1 17934	Value of the object part of the functor composition. (Contributed by Mario Carneiro, 28-Jan-2017.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    &   ⊢ (𝜑 → 𝐺 ∈ (𝐷 Func 𝐸))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ((1st ‘(𝐺 ∘func 𝐹))‘𝑋) = ((1st ‘𝐺)‘((1st ‘𝐹)‘𝑋)))
Theorem	cofu2nd 17935	Value of the morphism part of the functor composition. (Contributed by Mario Carneiro, 3-Jan-2017.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    &   ⊢ (𝜑 → 𝐺 ∈ (𝐷 Func 𝐸))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑋(2nd ‘(𝐺 ∘func 𝐹))𝑌) = ((((1st ‘𝐹)‘𝑋)(2nd ‘𝐺)((1st ‘𝐹)‘𝑌)) ∘ (𝑋(2nd ‘𝐹)𝑌)))
Theorem	cofu2 17936	Value of the morphism part of the functor composition. (Contributed by Mario Carneiro, 28-Jan-2017.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    &   ⊢ (𝜑 → 𝐺 ∈ (𝐷 Func 𝐸))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝑅 ∈ (𝑋𝐻𝑌))    ⇒   ⊢ (𝜑 → ((𝑋(2nd ‘(𝐺 ∘func 𝐹))𝑌)‘𝑅) = ((((1st ‘𝐹)‘𝑋)(2nd ‘𝐺)((1st ‘𝐹)‘𝑌))‘((𝑋(2nd ‘𝐹)𝑌)‘𝑅)))
Theorem	cofuval2 17937*	Value of the composition of two functors. (Contributed by Mario Carneiro, 3-Jan-2017.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)    &   ⊢ (𝜑 → 𝐻(𝐷 Func 𝐸)𝐾)    ⇒   ⊢ (𝜑 → (⟨𝐻, 𝐾⟩ ∘func ⟨𝐹, 𝐺⟩) = ⟨(𝐻 ∘ 𝐹), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (((𝐹‘𝑥)𝐾(𝐹‘𝑦)) ∘ (𝑥𝐺𝑦)))⟩)
Theorem	cofucl 17938	The composition of two functors is a functor. Proposition 3.23 of [Adamek] p. 33. (Contributed by Mario Carneiro, 3-Jan-2017.)
⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    &   ⊢ (𝜑 → 𝐺 ∈ (𝐷 Func 𝐸))    ⇒   ⊢ (𝜑 → (𝐺 ∘func 𝐹) ∈ (𝐶 Func 𝐸))
Theorem	cofuass 17939	Functor composition is associative. (Contributed by Mario Carneiro, 3-Jan-2017.)
⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐷))    &   ⊢ (𝜑 → 𝐻 ∈ (𝐷 Func 𝐸))    &   ⊢ (𝜑 → 𝐾 ∈ (𝐸 Func 𝐹))    ⇒   ⊢ (𝜑 → ((𝐾 ∘func 𝐻) ∘func 𝐺) = (𝐾 ∘func (𝐻 ∘func 𝐺)))
Theorem	cofulid 17940	The identity functor is a left identity for composition. (Contributed by Mario Carneiro, 3-Jan-2017.)
⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    &   ⊢ 𝐼 = (idfunc‘𝐷)    ⇒   ⊢ (𝜑 → (𝐼 ∘func 𝐹) = 𝐹)
Theorem	cofurid 17941	The identity functor is a right identity for composition. (Contributed by Mario Carneiro, 3-Jan-2017.)
⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    &   ⊢ 𝐼 = (idfunc‘𝐶)    ⇒   ⊢ (𝜑 → (𝐹 ∘func 𝐼) = 𝐹)
Theorem	resfval 17942*	Value of the functor restriction operator. (Contributed by Mario Carneiro, 6-Jan-2017.)
⊢ (𝜑 → 𝐹 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐻 ∈ 𝑊)    ⇒   ⊢ (𝜑 → (𝐹 ↾f 𝐻) = ⟨((1st ‘𝐹) ↾ dom dom 𝐻), (𝑥 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑥) ↾ (𝐻‘𝑥)))⟩)
Theorem	resfval2 17943*	Value of the functor restriction operator. (Contributed by Mario Carneiro, 6-Jan-2017.)
⊢ (𝜑 → 𝐹 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐻 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐻 Fn (𝑆 × 𝑆))    ⇒   ⊢ (𝜑 → (⟨𝐹, 𝐺⟩ ↾f 𝐻) = ⟨(𝐹 ↾ 𝑆), (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ ((𝑥𝐺𝑦) ↾ (𝑥𝐻𝑦)))⟩)
Theorem	resf1st 17944	Value of the functor restriction operator on objects. (Contributed by Mario Carneiro, 6-Jan-2017.)
⊢ (𝜑 → 𝐹 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐻 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐻 Fn (𝑆 × 𝑆))    ⇒   ⊢ (𝜑 → (1st ‘(𝐹 ↾f 𝐻)) = ((1st ‘𝐹) ↾ 𝑆))
Theorem	resf2nd 17945	Value of the functor restriction operator on morphisms. (Contributed by Mario Carneiro, 6-Jan-2017.)
⊢ (𝜑 → 𝐹 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐻 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐻 Fn (𝑆 × 𝑆))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑆)    ⇒   ⊢ (𝜑 → (𝑋(2nd ‘(𝐹 ↾f 𝐻))𝑌) = ((𝑋(2nd ‘𝐹)𝑌) ↾ (𝑋𝐻𝑌)))
Theorem	funcres 17946	A functor restricted to a subcategory is a functor. (Contributed by Mario Carneiro, 6-Jan-2017.)
⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    &   ⊢ (𝜑 → 𝐻 ∈ (Subcat‘𝐶))    ⇒   ⊢ (𝜑 → (𝐹 ↾f 𝐻) ∈ ((𝐶 ↾cat 𝐻) Func 𝐷))
Theorem	funcres2b 17947*	Condition for a functor to also be a functor into the restriction. (Contributed by Mario Carneiro, 6-Jan-2017.)
⊢ 𝐴 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝑅 ∈ (Subcat‘𝐷))    &   ⊢ (𝜑 → 𝑅 Fn (𝑆 × 𝑆))    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝑆)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥𝐺𝑦):𝑌⟶((𝐹‘𝑥)𝑅(𝐹‘𝑦)))    ⇒   ⊢ (𝜑 → (𝐹(𝐶 Func 𝐷)𝐺 ↔ 𝐹(𝐶 Func (𝐷 ↾cat 𝑅))𝐺))
Theorem	funcres2 17948	A functor into a restricted category is also a functor into the whole category. (Contributed by Mario Carneiro, 6-Jan-2017.)
⊢ (𝑅 ∈ (Subcat‘𝐷) → (𝐶 Func (𝐷 ↾cat 𝑅)) ⊆ (𝐶 Func 𝐷))
Theorem	idfusubc0 17949*	The identity functor for a subcategory is an "inclusion functor" from the subcategory into its supercategory. (Contributed by AV, 29-Mar-2020.)
⊢ 𝑆 = (𝐶 ↾cat 𝐽)    &   ⊢ 𝐼 = (idfunc‘𝑆)    &   ⊢ 𝐵 = (Base‘𝑆)    ⇒   ⊢ (𝐽 ∈ (Subcat‘𝐶) → 𝐼 = ⟨( I ↾ 𝐵), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥(Hom ‘𝑆)𝑦)))⟩)
Theorem	idfusubc 17950*	The identity functor for a subcategory is an "inclusion functor" from the subcategory into its supercategory. (Contributed by AV, 29-Mar-2020.)
⊢ 𝑆 = (𝐶 ↾cat 𝐽)    &   ⊢ 𝐼 = (idfunc‘𝑆)    &   ⊢ 𝐵 = (Base‘𝑆)    ⇒   ⊢ (𝐽 ∈ (Subcat‘𝐶) → 𝐼 = ⟨( I ↾ 𝐵), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥𝐽𝑦)))⟩)
Theorem	wunfunc 17951	A weak universe is closed under the functor set operation. (Contributed by Mario Carneiro, 12-Jan-2017.) (Proof shortened by AV, 13-Oct-2024.)
⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑈)    ⇒   ⊢ (𝜑 → (𝐶 Func 𝐷) ∈ 𝑈)
Theorem	wunfuncOLD 17952	Obsolete version of wunfunc 17951 as of 13-Oct-2024. (Contributed by Mario Carneiro, 12-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑈)    ⇒   ⊢ (𝜑 → (𝐶 Func 𝐷) ∈ 𝑈)
Theorem	funcpropd 17953	If two categories have the same set of objects, morphisms, and compositions, then they have the same functors. (Contributed by Mario Carneiro, 17-Jan-2017.)
⊢ (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 → (𝐴 Func 𝐶) = (𝐵 Func 𝐷))
Theorem	funcres2c 17954	Condition for a functor to also be a functor into the restriction. (Contributed by Mario Carneiro, 30-Jan-2017.)
⊢ 𝐴 = (Base‘𝐶)    &   ⊢ 𝐸 = (𝐷 ↾s 𝑆)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝑆)    ⇒   ⊢ (𝜑 → (𝐹(𝐶 Func 𝐷)𝐺 ↔ 𝐹(𝐶 Func 𝐸)𝐺))
8.1.8  Full & faithful functors
Syntax	cful 17955	Extend class notation with the class of all full functors.
class Full
Syntax	cfth 17956	Extend class notation with the class of all faithful functors.
class Faith
Definition	df-full 17957*	Function returning all the full functors from a category 𝐶 to a category 𝐷. A full functor is a functor in which all the morphism maps 𝐺(𝑋, 𝑌) between objects 𝑋, 𝑌 ∈ 𝐶 are surjections. Definition 3.27(3) in [Adamek] p. 34. (Contributed by Mario Carneiro, 26-Jan-2017.)
⊢ Full = (𝑐 ∈ Cat, 𝑑 ∈ Cat ↦ {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝑐 Func 𝑑)𝑔 ∧ ∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)ran (𝑥𝑔𝑦) = ((𝑓‘𝑥)(Hom ‘𝑑)(𝑓‘𝑦)))})
Definition	df-fth 17958*	Function returning all the faithful functors from a category 𝐶 to a category 𝐷. A faithful functor is a functor in which all the morphism maps 𝐺(𝑋, 𝑌) between objects 𝑋, 𝑌 ∈ 𝐶 are injections. Definition 3.27(2) in [Adamek] p. 34. (Contributed by Mario Carneiro, 26-Jan-2017.)
⊢ Faith = (𝑐 ∈ Cat, 𝑑 ∈ Cat ↦ {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝑐 Func 𝑑)𝑔 ∧ ∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)Fun ◡(𝑥𝑔𝑦))})
Theorem	fullfunc 17959	A full functor is a functor. (Contributed by Mario Carneiro, 26-Jan-2017.)
⊢ (𝐶 Full 𝐷) ⊆ (𝐶 Func 𝐷)
Theorem	fthfunc 17960	A faithful functor is a functor. (Contributed by Mario Carneiro, 26-Jan-2017.)
⊢ (𝐶 Faith 𝐷) ⊆ (𝐶 Func 𝐷)
Theorem	relfull 17961	The set of full functors is a relation. (Contributed by Mario Carneiro, 26-Jan-2017.)
⊢ Rel (𝐶 Full 𝐷)
Theorem	relfth 17962	The set of faithful functors is a relation. (Contributed by Mario Carneiro, 26-Jan-2017.)
⊢ Rel (𝐶 Faith 𝐷)
Theorem	isfull 17963*	Value of the set of full functors between two categories. (Contributed by Mario Carneiro, 27-Jan-2017.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐽 = (Hom ‘𝐷)    ⇒   ⊢ (𝐹(𝐶 Full 𝐷)𝐺 ↔ (𝐹(𝐶 Func 𝐷)𝐺 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ran (𝑥𝐺𝑦) = ((𝐹‘𝑥)𝐽(𝐹‘𝑦))))
Theorem	isfull2 17964*	Equivalent condition for a full functor. (Contributed by Mario Carneiro, 27-Jan-2017.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐽 = (Hom ‘𝐷)    &   ⊢ 𝐻 = (Hom ‘𝐶)    ⇒   ⊢ (𝐹(𝐶 Full 𝐷)𝐺 ↔ (𝐹(𝐶 Func 𝐷)𝐺 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)–onto→((𝐹‘𝑥)𝐽(𝐹‘𝑦))))
Theorem	fullfo 17965	The morphism map of a full functor is a surjection. (Contributed by Mario Carneiro, 27-Jan-2017.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐽 = (Hom ‘𝐷)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝐹(𝐶 Full 𝐷)𝐺)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑋𝐺𝑌):(𝑋𝐻𝑌)–onto→((𝐹‘𝑋)𝐽(𝐹‘𝑌)))
Theorem	fulli 17966*	The morphism map of a full functor is a surjection. (Contributed by Mario Carneiro, 27-Jan-2017.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐽 = (Hom ‘𝐷)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝐹(𝐶 Full 𝐷)𝐺)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑅 ∈ ((𝐹‘𝑋)𝐽(𝐹‘𝑌)))    ⇒   ⊢ (𝜑 → ∃𝑓 ∈ (𝑋𝐻𝑌)𝑅 = ((𝑋𝐺𝑌)‘𝑓))
Theorem	isfth 17967*	Value of the set of faithful functors between two categories. (Contributed by Mario Carneiro, 27-Jan-2017.)
⊢ 𝐵 = (Base‘𝐶)    ⇒   ⊢ (𝐹(𝐶 Faith 𝐷)𝐺 ↔ (𝐹(𝐶 Func 𝐷)𝐺 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 Fun ◡(𝑥𝐺𝑦)))
Theorem	isfth2 17968*	Equivalent condition for a faithful functor. (Contributed by Mario Carneiro, 27-Jan-2017.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ 𝐽 = (Hom ‘𝐷)    ⇒   ⊢ (𝐹(𝐶 Faith 𝐷)𝐺 ↔ (𝐹(𝐶 Func 𝐷)𝐺 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)–1-1→((𝐹‘𝑥)𝐽(𝐹‘𝑦))))
Theorem	isffth2 17969*	A fully faithful functor is a functor which is bijective on hom-sets. (Contributed by Mario Carneiro, 27-Jan-2017.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ 𝐽 = (Hom ‘𝐷)    ⇒   ⊢ (𝐹((𝐶 Full 𝐷) ∩ (𝐶 Faith 𝐷))𝐺 ↔ (𝐹(𝐶 Func 𝐷)𝐺 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)–1-1-onto→((𝐹‘𝑥)𝐽(𝐹‘𝑦))))
Theorem	fthf1 17970	The morphism map of a faithful functor is an injection. (Contributed by Mario Carneiro, 27-Jan-2017.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ 𝐽 = (Hom ‘𝐷)    &   ⊢ (𝜑 → 𝐹(𝐶 Faith 𝐷)𝐺)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑋𝐺𝑌):(𝑋𝐻𝑌)–1-1→((𝐹‘𝑋)𝐽(𝐹‘𝑌)))
Theorem	fthi 17971	The morphism map of a faithful functor is an injection. (Contributed by Mario Carneiro, 27-Jan-2017.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ 𝐽 = (Hom ‘𝐷)    &   ⊢ (𝜑 → 𝐹(𝐶 Faith 𝐷)𝐺)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑅 ∈ (𝑋𝐻𝑌))    &   ⊢ (𝜑 → 𝑆 ∈ (𝑋𝐻𝑌))    ⇒   ⊢ (𝜑 → (((𝑋𝐺𝑌)‘𝑅) = ((𝑋𝐺𝑌)‘𝑆) ↔ 𝑅 = 𝑆))
Theorem	ffthf1o 17972	The morphism map of a fully faithful functor is a bijection. (Contributed by Mario Carneiro, 29-Jan-2017.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ 𝐽 = (Hom ‘𝐷)    &   ⊢ (𝜑 → 𝐹((𝐶 Full 𝐷) ∩ (𝐶 Faith 𝐷))𝐺)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑋𝐺𝑌):(𝑋𝐻𝑌)–1-1-onto→((𝐹‘𝑋)𝐽(𝐹‘𝑌)))
Theorem	fullpropd 17973	If two categories have the same set of objects, morphisms, and compositions, then they have the same full functors. (Contributed by Mario Carneiro, 27-Jan-2017.)
⊢ (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 → (𝐴 Full 𝐶) = (𝐵 Full 𝐷))
Theorem	fthpropd 17974	If two categories have the same set of objects, morphisms, and compositions, then they have the same faithful functors. (Contributed by Mario Carneiro, 27-Jan-2017.)
⊢ (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 → (𝐴 Faith 𝐶) = (𝐵 Faith 𝐷))
Theorem	fulloppc 17975	The opposite functor of a full functor is also full. Proposition 3.43(d) in [Adamek] p. 39. (Contributed by Mario Carneiro, 27-Jan-2017.)
⊢ 𝑂 = (oppCat‘𝐶)    &   ⊢ 𝑃 = (oppCat‘𝐷)    &   ⊢ (𝜑 → 𝐹(𝐶 Full 𝐷)𝐺)    ⇒   ⊢ (𝜑 → 𝐹(𝑂 Full 𝑃)tpos 𝐺)
Theorem	fthoppc 17976	The opposite functor of a faithful functor is also faithful. Proposition 3.43(c) in [Adamek] p. 39. (Contributed by Mario Carneiro, 27-Jan-2017.)
⊢ 𝑂 = (oppCat‘𝐶)    &   ⊢ 𝑃 = (oppCat‘𝐷)    &   ⊢ (𝜑 → 𝐹(𝐶 Faith 𝐷)𝐺)    ⇒   ⊢ (𝜑 → 𝐹(𝑂 Faith 𝑃)tpos 𝐺)
Theorem	ffthoppc 17977	The opposite functor of a fully faithful functor is also full and faithful. (Contributed by Mario Carneiro, 27-Jan-2017.)
⊢ 𝑂 = (oppCat‘𝐶)    &   ⊢ 𝑃 = (oppCat‘𝐷)    &   ⊢ (𝜑 → 𝐹((𝐶 Full 𝐷) ∩ (𝐶 Faith 𝐷))𝐺)    ⇒   ⊢ (𝜑 → 𝐹((𝑂 Full 𝑃) ∩ (𝑂 Faith 𝑃))tpos 𝐺)
Theorem	fthsect 17978	A faithful functor reflects sections. (Contributed by Mario Carneiro, 27-Jan-2017.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝐹(𝐶 Faith 𝐷)𝐺)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑀 ∈ (𝑋𝐻𝑌))    &   ⊢ (𝜑 → 𝑁 ∈ (𝑌𝐻𝑋))    &   ⊢ 𝑆 = (Sect‘𝐶)    &   ⊢ 𝑇 = (Sect‘𝐷)    ⇒   ⊢ (𝜑 → (𝑀(𝑋𝑆𝑌)𝑁 ↔ ((𝑋𝐺𝑌)‘𝑀)((𝐹‘𝑋)𝑇(𝐹‘𝑌))((𝑌𝐺𝑋)‘𝑁)))
Theorem	fthinv 17979	A faithful functor reflects inverses. (Contributed by Mario Carneiro, 27-Jan-2017.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝐹(𝐶 Faith 𝐷)𝐺)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑀 ∈ (𝑋𝐻𝑌))    &   ⊢ (𝜑 → 𝑁 ∈ (𝑌𝐻𝑋))    &   ⊢ 𝐼 = (Inv‘𝐶)    &   ⊢ 𝐽 = (Inv‘𝐷)    ⇒   ⊢ (𝜑 → (𝑀(𝑋𝐼𝑌)𝑁 ↔ ((𝑋𝐺𝑌)‘𝑀)((𝐹‘𝑋)𝐽(𝐹‘𝑌))((𝑌𝐺𝑋)‘𝑁)))
Theorem	fthmon 17980	A faithful functor reflects monomorphisms. (Contributed by Mario Carneiro, 27-Jan-2017.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝐹(𝐶 Faith 𝐷)𝐺)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑅 ∈ (𝑋𝐻𝑌))    &   ⊢ 𝑀 = (Mono‘𝐶)    &   ⊢ 𝑁 = (Mono‘𝐷)    &   ⊢ (𝜑 → ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹‘𝑋)𝑁(𝐹‘𝑌)))    ⇒   ⊢ (𝜑 → 𝑅 ∈ (𝑋𝑀𝑌))
Theorem	fthepi 17981	A faithful functor reflects epimorphisms. (Contributed by Mario Carneiro, 27-Jan-2017.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝐹(𝐶 Faith 𝐷)𝐺)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑅 ∈ (𝑋𝐻𝑌))    &   ⊢ 𝐸 = (Epi‘𝐶)    &   ⊢ 𝑃 = (Epi‘𝐷)    &   ⊢ (𝜑 → ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹‘𝑋)𝑃(𝐹‘𝑌)))    ⇒   ⊢ (𝜑 → 𝑅 ∈ (𝑋𝐸𝑌))
Theorem	ffthiso 17982	A fully faithful functor reflects isomorphisms. Corollary 3.32 of [Adamek] p. 35. (Contributed by Mario Carneiro, 27-Jan-2017.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝐹(𝐶 Faith 𝐷)𝐺)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑅 ∈ (𝑋𝐻𝑌))    &   ⊢ (𝜑 → 𝐹(𝐶 Full 𝐷)𝐺)    &   ⊢ 𝐼 = (Iso‘𝐶)    &   ⊢ 𝐽 = (Iso‘𝐷)    ⇒   ⊢ (𝜑 → (𝑅 ∈ (𝑋𝐼𝑌) ↔ ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹‘𝑋)𝐽(𝐹‘𝑌))))
Theorem	fthres2b 17983*	Condition for a faithful functor to also be a faithful functor into the restriction. (Contributed by Mario Carneiro, 27-Jan-2017.)
⊢ 𝐴 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝑅 ∈ (Subcat‘𝐷))    &   ⊢ (𝜑 → 𝑅 Fn (𝑆 × 𝑆))    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝑆)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥𝐺𝑦):𝑌⟶((𝐹‘𝑥)𝑅(𝐹‘𝑦)))    ⇒   ⊢ (𝜑 → (𝐹(𝐶 Faith 𝐷)𝐺 ↔ 𝐹(𝐶 Faith (𝐷 ↾cat 𝑅))𝐺))
Theorem	fthres2c 17984	Condition for a faithful functor to also be a faithful functor into the restriction. (Contributed by Mario Carneiro, 30-Jan-2017.)
⊢ 𝐴 = (Base‘𝐶)    &   ⊢ 𝐸 = (𝐷 ↾s 𝑆)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝑆)    ⇒   ⊢ (𝜑 → (𝐹(𝐶 Faith 𝐷)𝐺 ↔ 𝐹(𝐶 Faith 𝐸)𝐺))
Theorem	fthres2 17985	A faithful functor into a restricted category is also a faithful functor into the whole category. (Contributed by Mario Carneiro, 27-Jan-2017.)
⊢ (𝑅 ∈ (Subcat‘𝐷) → (𝐶 Faith (𝐷 ↾cat 𝑅)) ⊆ (𝐶 Faith 𝐷))
Theorem	idffth 17986	The identity functor is a fully faithful functor. (Contributed by Mario Carneiro, 27-Jan-2017.)
⊢ 𝐼 = (idfunc‘𝐶)    ⇒   ⊢ (𝐶 ∈ Cat → 𝐼 ∈ ((𝐶 Full 𝐶) ∩ (𝐶 Faith 𝐶)))
Theorem	cofull 17987	The composition of two full functors is full. Proposition 3.30(d) in [Adamek] p. 35. (Contributed by Mario Carneiro, 28-Jan-2017.)
⊢ (𝜑 → 𝐹 ∈ (𝐶 Full 𝐷))    &   ⊢ (𝜑 → 𝐺 ∈ (𝐷 Full 𝐸))    ⇒   ⊢ (𝜑 → (𝐺 ∘func 𝐹) ∈ (𝐶 Full 𝐸))
Theorem	cofth 17988	The composition of two faithful functors is faithful. Proposition 3.30(c) in [Adamek] p. 35. (Contributed by Mario Carneiro, 28-Jan-2017.)
⊢ (𝜑 → 𝐹 ∈ (𝐶 Faith 𝐷))    &   ⊢ (𝜑 → 𝐺 ∈ (𝐷 Faith 𝐸))    ⇒   ⊢ (𝜑 → (𝐺 ∘func 𝐹) ∈ (𝐶 Faith 𝐸))
Theorem	coffth 17989	The composition of two fully faithful functors is fully faithful. (Contributed by Mario Carneiro, 28-Jan-2017.)
⊢ (𝜑 → 𝐹 ∈ ((𝐶 Full 𝐷) ∩ (𝐶 Faith 𝐷)))    &   ⊢ (𝜑 → 𝐺 ∈ ((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸)))    ⇒   ⊢ (𝜑 → (𝐺 ∘func 𝐹) ∈ ((𝐶 Full 𝐸) ∩ (𝐶 Faith 𝐸)))
Theorem	rescfth 17990	The inclusion functor from a subcategory is a faithful functor. (Contributed by Mario Carneiro, 27-Jan-2017.)
⊢ 𝐷 = (𝐶 ↾cat 𝐽)    &   ⊢ 𝐼 = (idfunc‘𝐷)    ⇒   ⊢ (𝐽 ∈ (Subcat‘𝐶) → 𝐼 ∈ (𝐷 Faith 𝐶))
Theorem	ressffth 17991	The inclusion functor from a full subcategory is a full and faithful functor, see also remark 4.4(2) in [Adamek] p. 49. (Contributed by Mario Carneiro, 27-Jan-2017.)
⊢ 𝐷 = (𝐶 ↾s 𝑆)    &   ⊢ 𝐼 = (idfunc‘𝐷)    ⇒   ⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → 𝐼 ∈ ((𝐷 Full 𝐶) ∩ (𝐷 Faith 𝐶)))
Theorem	fullres2c 17992	Condition for a full functor to also be a full functor into the restriction. (Contributed by Mario Carneiro, 30-Jan-2017.)
⊢ 𝐴 = (Base‘𝐶)    &   ⊢ 𝐸 = (𝐷 ↾s 𝑆)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝑆)    ⇒   ⊢ (𝜑 → (𝐹(𝐶 Full 𝐷)𝐺 ↔ 𝐹(𝐶 Full 𝐸)𝐺))
Theorem	ffthres2c 17993	Condition for a fully faithful functor to also be a fully faithful functor into the restriction. (Contributed by Mario Carneiro, 27-Jan-2017.)
⊢ 𝐴 = (Base‘𝐶)    &   ⊢ 𝐸 = (𝐷 ↾s 𝑆)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝑆)    ⇒   ⊢ (𝜑 → (𝐹((𝐶 Full 𝐷) ∩ (𝐶 Faith 𝐷))𝐺 ↔ 𝐹((𝐶 Full 𝐸) ∩ (𝐶 Faith 𝐸))𝐺))
Theorem	inclfusubc 17994*	The "inclusion functor" from a subcategory of a category into the category itself. (Contributed by AV, 30-Mar-2020.)
⊢ (𝜑 → 𝐽 ∈ (Subcat‘𝐶))    &   ⊢ 𝑆 = (𝐶 ↾cat 𝐽)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝐹 = ( I ↾ 𝐵))    &   ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥𝐽𝑦))))    ⇒   ⊢ (𝜑 → 𝐹(𝑆 Func 𝐶)𝐺)
8.1.9  Natural transformations and the functor category
Syntax	cnat 17995	Extend class notation to include the collection of natural transformations.
class Nat
Syntax	cfuc 17996	Extend class notation to include the functor category.
class FuncCat
Definition	df-nat 17997*	Definition of a natural transformation between two functors. A natural transformation 𝐴:𝐹⟶𝐺 is a collection of arrows 𝐴(𝑥):𝐹(𝑥)⟶𝐺(𝑥), such that 𝐴(𝑦) ∘ 𝐹(ℎ) = 𝐺(ℎ) ∘ 𝐴(𝑥) for each morphism ℎ:𝑥⟶𝑦. Definition 6.1 in [Adamek] p. 83, and definition in [Lang] p. 65. (Contributed by Mario Carneiro, 6-Jan-2017.)
⊢ Nat = (𝑡 ∈ Cat, 𝑢 ∈ Cat ↦ (𝑓 ∈ (𝑡 Func 𝑢), 𝑔 ∈ (𝑡 Func 𝑢) ↦ ⦋(1st ‘𝑓) / 𝑟⦌⦋(1st ‘𝑔) / 𝑠⦌{𝑎 ∈ X𝑥 ∈ (Base‘𝑡)((𝑟‘𝑥)(Hom ‘𝑢)(𝑠‘𝑥)) ∣ ∀𝑥 ∈ (Base‘𝑡)∀𝑦 ∈ (Base‘𝑡)∀ℎ ∈ (𝑥(Hom ‘𝑡)𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩(comp‘𝑢)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘ℎ)) = (((𝑥(2nd ‘𝑔)𝑦)‘ℎ)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩(comp‘𝑢)(𝑠‘𝑦))(𝑎‘𝑥))}))
Definition	df-fuc 17998*	Definition of the category of functors between two fixed categories, with the objects being functors and the morphisms being natural transformations. Definition 6.15 in [Adamek] p. 87. (Contributed by Mario Carneiro, 6-Jan-2017.)
⊢ FuncCat = (𝑡 ∈ Cat, 𝑢 ∈ Cat ↦ {⟨(Base‘ndx), (𝑡 Func 𝑢)⟩, ⟨(Hom ‘ndx), (𝑡 Nat 𝑢)⟩, ⟨(comp‘ndx), (𝑣 ∈ ((𝑡 Func 𝑢) × (𝑡 Func 𝑢)), ℎ ∈ (𝑡 Func 𝑢) ↦ ⦋(1st ‘𝑣) / 𝑓⦌⦋(2nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔(𝑡 Nat 𝑢)ℎ), 𝑎 ∈ (𝑓(𝑡 Nat 𝑢)𝑔) ↦ (𝑥 ∈ (Base‘𝑡) ↦ ((𝑏‘𝑥)(⟨((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)⟩(comp‘𝑢)((1st ‘ℎ)‘𝑥))(𝑎‘𝑥)))))⟩})
Theorem	fnfuc 17999	The FuncCat operation is a well-defined function on categories. (Contributed by Mario Carneiro, 12-Jan-2017.)
⊢ FuncCat Fn (Cat × Cat)
Theorem	natfval 18000*	Value of the function giving natural transformations between two categories. (Contributed by Mario Carneiro, 6-Jan-2017.) (Proof shortened by AV, 1-Mar-2024.)
⊢ 𝑁 = (𝐶 Nat 𝐷)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ 𝐽 = (Hom ‘𝐷)    &   ⊢ · = (comp‘𝐷)    ⇒   ⊢ 𝑁 = (𝑓 ∈ (𝐶 Func 𝐷), 𝑔 ∈ (𝐶 Func 𝐷) ↦ ⦋(1st ‘𝑓) / 𝑟⦌⦋(1st ‘𝑔) / 𝑠⦌{𝑎 ∈ X𝑥 ∈ 𝐵 ((𝑟‘𝑥)𝐽(𝑠‘𝑥)) ∣ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀ℎ ∈ (𝑥𝐻𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩ · (𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘ℎ)) = (((𝑥(2nd ‘𝑔)𝑦)‘ℎ)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩ · (𝑠‘𝑦))(𝑎‘𝑥))})