let A, B, C, D be Category; :: thesis: for F being Functor of A,B
for G1, G2 being Functor of B,C
for H being Functor of C,D
for t being natural_transformation of G1,G2 st G1 is_naturally_transformable_to G2 holds
(H * t) * F = H * (t * F)
let F be Functor of A,B; :: thesis: for G1, G2 being Functor of B,C
for H being Functor of C,D
for t being natural_transformation of G1,G2 st G1 is_naturally_transformable_to G2 holds
(H * t) * F = H * (t * F)
let G1, G2 be Functor of B,C; :: thesis: for H being Functor of C,D
for t being natural_transformation of G1,G2 st G1 is_naturally_transformable_to G2 holds
(H * t) * F = H * (t * F)
let H be Functor of C,D; :: thesis: for t being natural_transformation of G1,G2 st G1 is_naturally_transformable_to G2 holds
(H * t) * F = H * (t * F)
let t be natural_transformation of G1,G2; :: thesis: ( G1 is_naturally_transformable_to G2 implies (H * t) * F = H * (t * F) )
assume A1: G1 is_naturally_transformable_to G2 ; :: thesis: (H * t) * F = H * (t * F)
A2: H * (G1 * F) = (H * G1) * F by RELAT_1:36;
then reconsider v = H * (t * F) as natural_transformation of (H * G1) * F,(H * G2) * F by RELAT_1:36;
A3: H * (G2 * F) = (H * G2) * F by RELAT_1:36;

A4: now :: thesis: for a being Object of A holds ((H * t) * F) . a = v . a

let a be Object of A; :: thesis: ((H * t) * F) . a = v . a
A5: ( G1 . (F . a) = (G1 * F) . a & G2 . (F . a) = (G2 * F) . a ) by CAT_1:76;
thus ((H * t) * F) . a = (H * t) . (F . a) by A1, Th20, Th22
.= H /. (t . (F . a)) by A1, Th21
.= H /. ((t * F) . a) by A1, A5, Th22
.= v . a by A1, A2, A3, Th20, Th21 ; :: thesis: verum

end;

H * G1 is_naturally_transformable_to H * G2 by A1, Th20;
hence (H * t) * F = H * (t * F) by A4, Th20, Th24; :: thesis: verum