let C, D be Category; :: thesis: for T being Functor of C,D
for c, c' being Object of C st Hom c,c' <> {} holds
for f being Morphism of c,c' holds T . f in Hom (T . c),(T . c')
let T be Functor of C,D; :: thesis: for c, c' being Object of C st Hom c,c' <> {} holds
for f being Morphism of c,c' holds T . f in Hom (T . c),(T . c')
let c, c' be Object of C; :: thesis: ( Hom c,c' <> {} implies for f being Morphism of c,c' holds T . f in Hom (T . c),(T . c') )
assume A1: Hom c,c' <> {} ; :: thesis: for f being Morphism of c,c' holds T . f in Hom (T . c),(T . c')
let f be Morphism of c,c'; :: thesis: T . f in Hom (T . c),(T . c')
f in Hom c,c' by A1, Def7;
hence T . f in Hom (T . c),(T . c') by Th123; :: thesis: verum