let B, A be Category; :: thesis: for F being Functor of A,B
for a, b being Object of A st Hom a,b <> {} holds
for f being Morphism of a,b holds ((id F) . b) * (F . f) = (F . f) * ((id F) . a)
let F be Functor of A,B; :: thesis: for a, b being Object of A st Hom a,b <> {} holds
for f being Morphism of a,b holds ((id F) . b) * (F . f) = (F . f) * ((id F) . a)
let a, b be Object of A; :: thesis: ( Hom a,b <> {} implies for f being Morphism of a,b holds ((id F) . b) * (F . f) = (F . f) * ((id F) . a) )
assume A1: Hom a,b <> {} ; :: thesis: for f being Morphism of a,b holds ((id F) . b) * (F . f) = (F . f) * ((id F) . a)
A2: ( Hom a,a <> {} & Hom b,b <> {} ) by CAT_1:56;
let f be Morphism of a,b; :: thesis: ((id F) . b) * (F . f) = (F . f) * ((id F) . a)
thus ((id F) . b) * (F . f) = (id (F . b)) * (F . f) by Th21
.= (F . (id b)) * (F . f) by Th15
.= F . ((id b) * f) by A1, A2, Th13
.= F . f by A1, CAT_1:57
.= F . (f * (id a)) by A1, CAT_1:58
.= (F . f) * (F . (id a)) by A1, A2, Th13
.= (F . f) * (id (F . a)) by Th15
.= (F . f) * ((id F) . a) by Th21 ; :: thesis: verum