let C, D be Category; :: thesis: for T being Functor of C,D
for c, c9 being Object of C st Hom c,c9 <> {} holds
for f being Morphism of c,c9 holds (hom T,c,c9) . f = T . f
let T be Functor of C,D; :: thesis: for c, c9 being Object of C st Hom c,c9 <> {} holds
for f being Morphism of c,c9 holds (hom T,c,c9) . f = T . f
let c, c9 be Object of C; :: thesis: ( Hom c,c9 <> {} implies for f being Morphism of c,c9 holds (hom T,c,c9) . f = T . f )
assume A1: Hom c,c9 <> {} ; :: thesis: for f being Morphism of c,c9 holds (hom T,c,c9) . f = T . f
let f be Morphism of c,c9; :: thesis: (hom T,c,c9) . f = T . f
f in Hom c,c9 by A1, Def7;
hence (hom T,c,c9) . f = T . f by FUNCT_1:72; :: thesis: verum