let A, B be non empty transitive with_units AltCatStr ; for F being covariant Functor of A,B
for o1, o2 being Object of A
for a being Morphism of o1,o2 st F is full & F is faithful & <^o1,o2^> <> {} & <^o2,o1^> <> {} & F . a is retraction holds
a is retraction
let F be covariant Functor of A,B; for o1, o2 being Object of A
for a being Morphism of o1,o2 st F is full & F is faithful & <^o1,o2^> <> {} & <^o2,o1^> <> {} & F . a is retraction holds
a is retraction
let o1, o2 be Object of A; for a being Morphism of o1,o2 st F is full & F is faithful & <^o1,o2^> <> {} & <^o2,o1^> <> {} & F . a is retraction holds
a is retraction
let a be Morphism of o1,o2; ( F is full & F is faithful & <^o1,o2^> <> {} & <^o2,o1^> <> {} & F . a is retraction implies a is retraction )
assume that
A1:
( F is full & F is faithful )
and
A2:
<^o1,o2^> <> {}
and
A3:
<^o2,o1^> <> {}
; ( not F . a is retraction or a is retraction )
A4:
<^(F . o2),(F . o1)^> <> {}
by A3, FUNCTOR0:def 18;
assume
F . a is retraction
; a is retraction
then consider b being Morphism of (F . o2),(F . o1) such that
A5:
b is_right_inverse_of F . a
;
Morph-Map (F,o2,o1) is onto
by A1, FUNCTOR1:15;
then
rng (Morph-Map (F,o2,o1)) = <^(F . o2),(F . o1)^>
;
then consider a9 being object such that
A6:
a9 in dom (Morph-Map (F,o2,o1))
and
A7:
b = (Morph-Map (F,o2,o1)) . a9
by A4, FUNCT_1:def 3;
reconsider a9 = a9 as Morphism of o2,o1 by A4, A6, FUNCT_2:def 1;
take
a9
; ALTCAT_3:def 2 a is_left_inverse_of a9
A8:
(F . a) * b = idm (F . o2)
by A5;
A9:
( dom (Morph-Map (F,o2,o2)) = <^o2,o2^> & Morph-Map (F,o2,o2) is one-to-one )
by A1, FUNCTOR1:16, FUNCT_2:def 1;
(Morph-Map (F,o2,o2)) . (idm o2) =
F . (idm o2)
by FUNCTOR0:def 15
.=
idm (F . o2)
by FUNCTOR2:1
.=
(F . a) * (F . a9)
by A3, A8, A4, A7, FUNCTOR0:def 15
.=
F . (a * a9)
by A2, A3, FUNCTOR0:def 23
.=
(Morph-Map (F,o2,o2)) . (a * a9)
by FUNCTOR0:def 15
;
hence
a * a9 = idm o2
by A9, FUNCT_1:def 4; ALTCAT_3:def 1 verum