let A, B be category; for F, G being covariant Functor of A,B st ( for a being Object of A holds F . a = G . a ) & ( for a, b being Object of A st <^a,b^> <> {} holds
for f being Morphism of a,b holds F . f = G . f ) holds
FunctorStr(# the ObjectMap of F, the MorphMap of F #) = FunctorStr(# the ObjectMap of G, the MorphMap of G #)
let F, G be covariant Functor of A,B; ( ( for a being Object of A holds F . a = G . a ) & ( for a, b being Object of A st <^a,b^> <> {} holds
for f being Morphism of a,b holds F . f = G . f ) implies FunctorStr(# the ObjectMap of F, the MorphMap of F #) = FunctorStr(# the ObjectMap of G, the MorphMap of G #) )
assume that
A1:
for a being Object of A holds F . a = G . a
and
A2:
for a, b being Object of A st <^a,b^> <> {} holds
for f being Morphism of a,b holds F . f = G . f
; FunctorStr(# the ObjectMap of F, the MorphMap of F #) = FunctorStr(# the ObjectMap of G, the MorphMap of G #)
the ObjectMap of F is Covariant
by FUNCTOR0:def 12;
then consider ff being Function of the carrier of A, the carrier of B such that
A3:
the ObjectMap of F = [:ff,ff:]
;
the ObjectMap of G is Covariant
by FUNCTOR0:def 12;
then consider gg being Function of the carrier of A, the carrier of B such that
A4:
the ObjectMap of G = [:gg,gg:]
;
now for a, b being Element of A holds the ObjectMap of F . (a,b) = the ObjectMap of G . (a,b)let a,
b be
Element of
A;
the ObjectMap of F . (a,b) = the ObjectMap of G . (a,b)reconsider x =
a,
y =
b as
Object of
A ;
A5:
dom ff = the
carrier of
A
by FUNCT_2:def 1;
A6:
dom gg = the
carrier of
A
by FUNCT_2:def 1;
A7:
the
ObjectMap of
F . (
x,
x)
= [(ff . x),(ff . x)]
by A3, A5, FUNCT_3:def 8;
A8:
the
ObjectMap of
F . (
y,
y)
= [(ff . y),(ff . y)]
by A3, A5, FUNCT_3:def 8;
A9:
the
ObjectMap of
G . (
x,
x)
= [(gg . x),(gg . x)]
by A4, A6, FUNCT_3:def 8;
A10:
the
ObjectMap of
G . (
y,
y)
= [(gg . y),(gg . y)]
by A4, A6, FUNCT_3:def 8;
A11:
F . x = ff . x
by A7;
A12:
F . y = ff . y
by A8;
A13:
G . x = gg . x
by A9;
A14:
G . y = gg . y
by A10;
A15:
F . x = G . x
by A1;
A16:
F . y = G . y
by A1;
thus the
ObjectMap of
F . (
a,
b) =
[(ff . a),(ff . b)]
by A3, A5, FUNCT_3:def 8
.=
the
ObjectMap of
G . (
a,
b)
by A4, A6, A11, A12, A13, A14, A15, A16, FUNCT_3:def 8
;
verum end;
then A17:
the ObjectMap of F = the ObjectMap of G
;
now for i being object st i in [: the carrier of A, the carrier of A:] holds
the MorphMap of F . i = the MorphMap of G . ilet i be
object ;
( i in [: the carrier of A, the carrier of A:] implies the MorphMap of F . i = the MorphMap of G . i )assume
i in [: the carrier of A, the carrier of A:]
;
the MorphMap of F . i = the MorphMap of G . ithen consider a,
b being
object such that A18:
a in the
carrier of
A
and A19:
b in the
carrier of
A
and A20:
i = [a,b]
by ZFMISC_1:def 2;
reconsider x =
a,
y =
b as
Object of
A by A18, A19;
A21:
(
<^x,y^> <> {} implies
<^(F . x),(F . y)^> <> {} )
by FUNCTOR0:def 18;
A22:
(
<^x,y^> <> {} implies
<^(G . x),(G . y)^> <> {} )
by FUNCTOR0:def 18;
A23:
dom (Morph-Map (F,x,y)) = <^x,y^>
by A21, FUNCT_2:def 1;
A24:
dom (Morph-Map (G,x,y)) = <^x,y^>
by A22, FUNCT_2:def 1;
now for z being object st z in <^x,y^> holds
(Morph-Map (F,x,y)) . z = (Morph-Map (G,x,y)) . zlet z be
object ;
( z in <^x,y^> implies (Morph-Map (F,x,y)) . z = (Morph-Map (G,x,y)) . z )assume A25:
z in <^x,y^>
;
(Morph-Map (F,x,y)) . z = (Morph-Map (G,x,y)) . zthen reconsider f =
z as
Morphism of
x,
y ;
thus (Morph-Map (F,x,y)) . z =
F . f
by A21, A25, FUNCTOR0:def 15
.=
G . f
by A2, A25
.=
(Morph-Map (G,x,y)) . z
by A22, A25, FUNCTOR0:def 15
;
verum end; hence
the
MorphMap of
F . i = the
MorphMap of
G . i
by A20, A23, A24;
verum end;
hence
FunctorStr(# the ObjectMap of F, the MorphMap of F #) = FunctorStr(# the ObjectMap of G, the MorphMap of G #)
by A17, PBOOLE:3; verum