let A, B, C be Category; :: thesis: for F1, F2 being Functor of A,B st F1 is_naturally_transformable_to F2 holds
for t being natural_transformation of F1,F2
for G being Functor of B,C
for a being Object of A holds (G * t) . a = G /. (t . a)
let F1, F2 be Functor of A,B; :: thesis: ( F1 is_naturally_transformable_to F2 implies for t being natural_transformation of F1,F2
for G being Functor of B,C
for a being Object of A holds (G * t) . a = G /. (t . a) )
assume A1: F1 is_naturally_transformable_to F2 ; :: thesis: for t being natural_transformation of F1,F2
for G being Functor of B,C
for a being Object of A holds (G * t) . a = G /. (t . a)
then A2: F1 is_transformable_to F2 ;
let t be natural_transformation of F1,F2; :: thesis: for G being Functor of B,C
for a being Object of A holds (G * t) . a = G /. (t . a)
let G be Functor of B,C; :: thesis: for a being Object of A holds (G * t) . a = G /. (t . a)
let a be Object of A; :: thesis: (G * t) . a = G /. (t . a)
thus (G * t) . a = (G * t) . a by A1, Def7
.= G /. (t . a) by A2, Th19 ; :: thesis: verum