let B, C, A be Category; :: thesis: for G1, G2 being Functor of B,C st G1 is_naturally_transformable_to G2 holds
for F being Functor of A,B
for t being natural_transformation of G1,G2
for a being Object of A holds (t * F) . a = t . (F . a)
let G1, G2 be Functor of B,C; :: thesis: ( G1 is_naturally_transformable_to G2 implies for F being Functor of A,B
for t being natural_transformation of G1,G2
for a being Object of A holds (t * F) . a = t . (F . a) )
assume A1: G1 is_naturally_transformable_to G2 ; :: thesis: for F being Functor of A,B
for t being natural_transformation of G1,G2
for a being Object of A holds (t * F) . a = t . (F . a)
then A2: G1 is_transformable_to G2 by NATTRA_1:def 7;
let F be Functor of A,B; :: thesis: for t being natural_transformation of G1,G2
for a being Object of A holds (t * F) . a = t . (F . a)
let t be natural_transformation of G1,G2; :: thesis: for a being Object of A holds (t * F) . a = t . (F . a)
let a be Object of A; :: thesis: (t * F) . a = t . (F . a)
thus (t * F) . a = (t * F) . a by A1, Def8
.= t . (F . a) by A2, Th25 ; :: thesis: verum