let C, D be Category; :: thesis: for T being Functor of C,D
for c, c9 being Object of C
for f being set st f in Hom c,c9 holds
T . f in Hom (T . c),(T . c9)
let T be Functor of C,D; :: thesis: for c, c9 being Object of C
for f being set st f in Hom c,c9 holds
T . f in Hom (T . c),(T . c9)
let c, c9 be Object of C; :: thesis: for f being set st f in Hom c,c9 holds
T . f in Hom (T . c),(T . c9)
let f be set ; :: thesis: ( f in Hom c,c9 implies T . f in Hom (T . c),(T . c9) )
assume A1: f in Hom c,c9 ; :: thesis: T . f in Hom (T . c),(T . c9)
then reconsider f9 = f as Morphism of c,c9 by Def7;
cod f9 = c9 by A1, Th18;
then A2: T . c9 = cod (T . f9) by Th105;
dom f9 = c by A1, Th18;
then T . c = dom (T . f9) by Th105;
hence T . f in Hom (T . c),(T . c9) by A2; :: thesis: verum