let A be non empty AltCatStr ; for B, C being non empty reflexive AltCatStr
for F being feasible Covariant FunctorStr over A,B
for N being feasible Contravariant FunctorStr over B,C
for o1, o2 being Object of A
for m being Morphism of o1,o2 st <^o1,o2^> <> {} holds
(N * F) . m = N . (F . m)
let B, C be non empty reflexive AltCatStr ; for F being feasible Covariant FunctorStr over A,B
for N being feasible Contravariant FunctorStr over B,C
for o1, o2 being Object of A
for m being Morphism of o1,o2 st <^o1,o2^> <> {} holds
(N * F) . m = N . (F . m)
let F be feasible Covariant FunctorStr over A,B; for N being feasible Contravariant FunctorStr over B,C
for o1, o2 being Object of A
for m being Morphism of o1,o2 st <^o1,o2^> <> {} holds
(N * F) . m = N . (F . m)
let N be feasible Contravariant FunctorStr over B,C; for o1, o2 being Object of A
for m being Morphism of o1,o2 st <^o1,o2^> <> {} holds
(N * F) . m = N . (F . m)
let o1, o2 be Object of A; for m being Morphism of o1,o2 st <^o1,o2^> <> {} holds
(N * F) . m = N . (F . m)
let m be Morphism of o1,o2; ( <^o1,o2^> <> {} implies (N * F) . m = N . (F . m) )
set I = the carrier of A;
reconsider s = the MorphMap of F . (o1,o2) as Function ;
reconsider r = (( the MorphMap of N * the ObjectMap of F) ** the MorphMap of F) . (o1,o2) as Function ;
reconsider t = ( the MorphMap of N * the ObjectMap of F) . (o1,o2) as Function ;
A1: dom (( the MorphMap of N * the ObjectMap of F) ** the MorphMap of F) =
(dom ( the MorphMap of N * the ObjectMap of F)) /\ (dom the MorphMap of F)
by PBOOLE:def 19
.=
[: the carrier of A, the carrier of A:] /\ (dom the MorphMap of F)
by PARTFUN1:def 2
.=
[: the carrier of A, the carrier of A:] /\ [: the carrier of A, the carrier of A:]
by PARTFUN1:def 2
.=
[: the carrier of A, the carrier of A:]
;
A2:
dom the ObjectMap of F = [: the carrier of A, the carrier of A:]
by FUNCT_2:def 1;
A3:
[o1,o2] in [: the carrier of A, the carrier of A:]
by ZFMISC_1:def 2;
assume A4:
<^o1,o2^> <> {}
; (N * F) . m = N . (F . m)
then A5:
<^(F . o1),(F . o2)^> <> {}
by FUNCTOR0:def 18;
then A6:
dom (Morph-Map (F,o1,o2)) = <^o1,o2^>
by FUNCT_2:def 1;
A7:
<^(N . (F . o2)),(N . (F . o1))^> <> {}
by A5, FUNCTOR0:def 19;
( (N * F) . o1 = N . (F . o1) & (N * F) . o2 = N . (F . o2) )
by FUNCTOR0:33;
hence (N * F) . m =
(Morph-Map ((N * F),o1,o2)) . m
by A4, A7, FUNCTOR0:def 16
.=
r . m
by FUNCTOR0:def 36
.=
(t * s) . m
by A1, A3, PBOOLE:def 19
.=
t . ((Morph-Map (F,o1,o2)) . m)
by A4, A6, FUNCT_1:13
.=
t . (F . m)
by A4, A5, FUNCTOR0:def 15
.=
( the MorphMap of N . ( the ObjectMap of F . (o1,o2))) . (F . m)
by A2, A3, FUNCT_1:13
.=
(Morph-Map (N,(F . o1),(F . o2))) . (F . m)
by FUNCTOR0:22
.=
N . (F . m)
by A5, A7, FUNCTOR0:def 16
;
verum