let A, B, C be Category; for F being Functor of A,B
for G being Functor of B,C
for a, b being Object of A st Hom (a,b) <> {} holds
for f being Morphism of a,b holds (G * F) /. f = G /. (F /. f)
let F be Functor of A,B; for G being Functor of B,C
for a, b being Object of A st Hom (a,b) <> {} holds
for f being Morphism of a,b holds (G * F) /. f = G /. (F /. f)
let G be Functor of B,C; for a, b being Object of A st Hom (a,b) <> {} holds
for f being Morphism of a,b holds (G * F) /. f = G /. (F /. f)
let a, b be Object of A; ( Hom (a,b) <> {} implies for f being Morphism of a,b holds (G * F) /. f = G /. (F /. f) )
assume A1:
Hom (a,b) <> {}
; for f being Morphism of a,b holds (G * F) /. f = G /. (F /. f)
then A2:
Hom ((F . a),(F . b)) <> {}
by CAT_1:84;
let f be Morphism of a,b; (G * F) /. f = G /. (F /. f)
thus (G * F) /. f =
(G * F) . f
by A1, CAT_3:def 10
.=
G . (F . f)
by FUNCT_2:15
.=
G . (F /. f)
by A1, CAT_3:def 10
.=
G /. (F /. f)
by A2, CAT_3:def 10
; verum