let A be Category; for f being Morphism of A holds <|(cod f),?> is_naturally_transformable_to <|(dom f),?>
let f be Morphism of A; <|(cod f),?> is_naturally_transformable_to <|(dom f),?>
set F1 = <|(cod f),?>;
set F2 = <|(dom f),?>;
set B = EnsHom A;
deffunc H1( Element of A) -> object = [[(Hom ((cod f),$1)),(Hom ((dom f),$1))],(hom (f,$1))];
A1:
for a being Object of A holds [[(Hom ((cod f),a)),(Hom ((dom f),a))],(hom (f,a))] in Hom ((<|(cod f),?> . a),(<|(dom f),?> . a))
proof
let a be
Object of
A;
[[(Hom ((cod f),a)),(Hom ((dom f),a))],(hom (f,a))] in Hom ((<|(cod f),?> . a),(<|(dom f),?> . a))
A2:
EnsHom A = CatStr(#
(Hom A),
(Maps (Hom A)),
(fDom (Hom A)),
(fCod (Hom A)),
(fComp (Hom A)) #)
by ENS_1:def 13;
then reconsider m =
[[(Hom ((cod f),a)),(Hom ((dom f),a))],(hom (f,a))] as
Morphism of
(EnsHom A) by ENS_1:48;
reconsider m9 =
m as
Element of
Maps (Hom A) by ENS_1:48;
A3:
cod m =
(fCod (Hom A)) . m
by A2
.=
cod m9
by ENS_1:def 10
.=
(m `1) `2
by ENS_1:def 4
.=
[(Hom ((cod f),a)),(Hom ((dom f),a))] `2
.=
Hom (
(dom f),
a)
.=
(Obj (hom?- ((Hom A),(dom f)))) . a
by ENS_1:60
.=
(hom?- ((Hom A),(dom f))) . a
.=
<|(dom f),?> . a
by ENS_1:def 25
;
dom m =
(fDom (Hom A)) . m
by A2
.=
dom m9
by ENS_1:def 9
.=
(m `1) `1
by ENS_1:def 3
.=
[(Hom ((cod f),a)),(Hom ((dom f),a))] `1
.=
Hom (
(cod f),
a)
.=
(Obj (hom?- ((Hom A),(cod f)))) . a
by ENS_1:60
.=
(hom?- ((Hom A),(cod f))) . a
.=
<|(cod f),?> . a
by ENS_1:def 25
;
hence
[[(Hom ((cod f),a)),(Hom ((dom f),a))],(hom (f,a))] in Hom (
(<|(cod f),?> . a),
(<|(dom f),?> . a))
by A3;
verum
end;
A4:
for a being Element of A holds H1(a) in the carrier' of (EnsHom A)
consider t being Function of the carrier of A, the carrier' of (EnsHom A) such that
A5:
for o being Element of A holds t . o = H1(o)
from FUNCT_2:sch 8(A4);
A6:
for a being Object of A holds t . a is Morphism of <|(cod f),?> . a,<|(dom f),?> . a
proof
let a be
Object of
A;
t . a is Morphism of <|(cod f),?> . a,<|(dom f),?> . a
[[(Hom ((cod f),a)),(Hom ((dom f),a))],(hom (f,a))] in Hom (
(<|(cod f),?> . a),
(<|(dom f),?> . a))
by A1;
then
[[(Hom ((cod f),a)),(Hom ((dom f),a))],(hom (f,a))] is
Morphism of
<|(cod f),?> . a,
<|(dom f),?> . a
by CAT_1:def 5;
hence
t . a is
Morphism of
<|(cod f),?> . a,
<|(dom f),?> . a
by A5;
verum
end;
for a being Object of A holds Hom ((<|(cod f),?> . a),(<|(dom f),?> . a)) <> {}
by A1;
then A7:
<|(cod f),?> is_transformable_to <|(dom f),?>
by NATTRA_1:def 2;
then reconsider t = t as transformation of <|(cod f),?>,<|(dom f),?> by A6, NATTRA_1:def 3;
for a, b being Object of A st Hom (a,b) <> {} holds
for g being Morphism of a,b holds (t . b) * (<|(cod f),?> /. g) = (<|(dom f),?> /. g) * (t . a)
proof
let a,
b be
Object of
A;
( Hom (a,b) <> {} implies for g being Morphism of a,b holds (t . b) * (<|(cod f),?> /. g) = (<|(dom f),?> /. g) * (t . a) )
assume A8:
Hom (
a,
b)
<> {}
;
for g being Morphism of a,b holds (t . b) * (<|(cod f),?> /. g) = (<|(dom f),?> /. g) * (t . a)
A9:
Hom (
(<|(cod f),?> . a),
(<|(cod f),?> . b))
<> {}
by A8, CAT_1:84;
let g be
Morphism of
a,
b;
(t . b) * (<|(cod f),?> /. g) = (<|(dom f),?> /. g) * (t . a)
A10:
dom g = a
by A8, CAT_1:5;
A11:
rng (hom ((cod f),g)) c= dom (hom (f,b))
proof
A12:
cod g = b
by A8, CAT_1:5;
per cases
( Hom ((dom f),b) = {} or Hom ((dom f),b) <> {} )
;
suppose A14:
Hom (
(dom f),
b)
<> {}
;
rng (hom ((cod f),g)) c= dom (hom (f,b))
cod g = b
by A8, CAT_1:5;
then A15:
(
rng (hom ((cod f),g)) c= Hom (
(cod f),
(cod g)) &
Hom (
(cod f),
(cod g))
= dom (hom (f,b)) )
by A14, FUNCT_2:def 1, RELAT_1:def 19;
let e be
object ;
TARSKI:def 3 ( not e in rng (hom ((cod f),g)) or e in dom (hom (f,b)) )assume
e in rng (hom ((cod f),g))
;
e in dom (hom (f,b))hence
e in dom (hom (f,b))
by A15;
verum end;
end;
A16:
rng (hom (f,a)) c= dom (hom ((dom f),g))
proof
A17:
dom g = a
by A8, CAT_1:5;
per cases
( Hom ((dom f),(cod g)) = {} or Hom ((dom f),(cod g)) <> {} )
;
suppose A19:
Hom (
(dom f),
(cod g))
<> {}
;
rng (hom (f,a)) c= dom (hom ((dom f),g))let e be
object ;
TARSKI:def 3 ( not e in rng (hom (f,a)) or e in dom (hom ((dom f),g)) )assume A20:
e in rng (hom (f,a))
;
e in dom (hom ((dom f),g))
(
rng (hom (f,a)) c= Hom (
(dom f),
a) &
Hom (
(dom f),
a)
= dom (hom ((dom f),g)) )
by A17, A19, FUNCT_2:def 1, RELAT_1:def 19;
hence
e in dom (hom ((dom f),g))
by A20;
verum end;
end;
A21:
dom ((hom (f,b)) * (hom ((cod f),g))) = dom ((hom ((dom f),g)) * (hom (f,a)))
proof
per cases
( Hom ((cod f),(dom g)) = {} or Hom ((cod f),(dom g)) <> {} )
;
suppose A22:
Hom (
(cod f),
(dom g))
= {}
;
dom ((hom (f,b)) * (hom ((cod f),g))) = dom ((hom ((dom f),g)) * (hom (f,a))) dom ((hom (f,b)) * (hom ((cod f),g))) =
dom (hom ((cod f),g))
by A11, RELAT_1:27
.=
Hom (
(cod f),
(dom g))
by A22
.=
dom (hom (f,a))
by A10, A22
.=
dom ((hom ((dom f),g)) * (hom (f,a)))
by A16, RELAT_1:27
;
hence
dom ((hom (f,b)) * (hom ((cod f),g))) = dom ((hom ((dom f),g)) * (hom (f,a)))
;
verum end; suppose A23:
Hom (
(cod f),
(dom g))
<> {}
;
dom ((hom (f,b)) * (hom ((cod f),g))) = dom ((hom ((dom f),g)) * (hom (f,a)))then A24:
Hom (
(cod f),
(cod g))
<> {}
by ENS_1:42;
A25:
Hom (
(dom f),
a)
<> {}
by A10, A23, ENS_1:42;
dom ((hom (f,b)) * (hom ((cod f),g))) =
dom (hom ((cod f),g))
by A11, RELAT_1:27
.=
Hom (
(cod f),
(dom g))
by A24, FUNCT_2:def 1
.=
Hom (
(cod f),
a)
by A8, CAT_1:5
.=
dom (hom (f,a))
by A25, FUNCT_2:def 1
.=
dom ((hom ((dom f),g)) * (hom (f,a)))
by A16, RELAT_1:27
;
hence
dom ((hom (f,b)) * (hom ((cod f),g))) = dom ((hom ((dom f),g)) * (hom (f,a)))
;
verum end;
end;
A26:
for
x being
object st
x in dom ((hom (f,b)) * (hom ((cod f),g))) holds
((hom (f,b)) * (hom ((cod f),g))) . x = ((hom ((dom f),g)) * (hom (f,a))) . x
proof
let x be
object ;
( x in dom ((hom (f,b)) * (hom ((cod f),g))) implies ((hom (f,b)) * (hom ((cod f),g))) . x = ((hom ((dom f),g)) * (hom (f,a))) . x )
assume A27:
x in dom ((hom (f,b)) * (hom ((cod f),g)))
;
((hom (f,b)) * (hom ((cod f),g))) . x = ((hom ((dom f),g)) * (hom (f,a))) . x
per cases
( Hom ((cod f),(dom g)) <> {} or Hom ((cod f),(dom g)) = {} )
;
suppose A28:
Hom (
(cod f),
(dom g))
<> {}
;
((hom (f,b)) * (hom ((cod f),g))) . x = ((hom ((dom f),g)) * (hom (f,a))) . xA29:
x in dom (hom ((cod f),g))
by A27, FUNCT_1:11;
Hom (
(cod f),
(cod g))
<> {}
by A28, ENS_1:42;
then A30:
x in Hom (
(cod f),
(dom g))
by A29, FUNCT_2:def 1;
then reconsider x =
x as
Morphism of
A ;
A31:
(
dom g = cod x &
dom x = cod f )
by A30, CAT_1:1;
A32:
dom g = cod x
by A30, CAT_1:1;
then A33:
cod (g (*) x) =
cod g
by CAT_1:17
.=
b
by A8, CAT_1:5
;
A34:
(hom (f,a)) . x = x (*) f
by A10, A30, ENS_1:def 20;
then reconsider h =
(hom (f,a)) . x as
Morphism of
A ;
A35:
dom x = cod f
by A30, CAT_1:1;
then A36:
dom (x (*) f) = dom f
by CAT_1:17;
dom (g (*) x) =
dom x
by A32, CAT_1:17
.=
cod f
by A30, CAT_1:1
;
then A37:
g (*) x in Hom (
(cod f),
b)
by A33;
cod (x (*) f) =
cod x
by A35, CAT_1:17
.=
dom g
by A30, CAT_1:1
;
then A38:
(hom (f,a)) . x in Hom (
(dom f),
(dom g))
by A34, A36;
((hom (f,b)) * (hom ((cod f),g))) . x =
(hom (f,b)) . ((hom ((cod f),g)) . x)
by A27, FUNCT_1:12
.=
(hom (f,b)) . (g (*) x)
by A30, ENS_1:def 19
.=
(g (*) x) (*) f
by A37, ENS_1:def 20
.=
g (*) (x (*) f)
by A31, CAT_1:18
.=
g (*) h
by A10, A30, ENS_1:def 20
.=
(hom ((dom f),g)) . ((hom (f,a)) . x)
by A38, ENS_1:def 19
.=
((hom ((dom f),g)) * (hom (f,a))) . x
by A21, A27, FUNCT_1:12
;
hence
((hom (f,b)) * (hom ((cod f),g))) . x = ((hom ((dom f),g)) * (hom (f,a))) . x
;
verum end;
end;
A40:
Hom (
(<|(dom f),?> . a),
(<|(dom f),?> . b))
<> {}
by A8, CAT_1:84;
A41:
cod g = b
by A8, CAT_1:5;
reconsider f4 =
t . a as
Morphism of
(EnsHom A) ;
A42:
t . a =
t . a
by A7, NATTRA_1:def 5
.=
[[(Hom ((cod f),a)),(Hom ((dom f),a))],(hom (f,a))]
by A5
;
then reconsider f49 =
f4 as
Element of
Maps (Hom A) by ENS_1:48;
A43:
Hom (
(<|(cod f),?> . a),
(<|(dom f),?> . a))
<> {}
by A1;
reconsider f1 =
t . b as
Morphism of
(EnsHom A) ;
A44:
t . b =
t . b
by A7, NATTRA_1:def 5
.=
[[(Hom ((cod f),b)),(Hom ((dom f),b))],(hom (f,b))]
by A5
;
then reconsider f19 =
f1 as
Element of
Maps (Hom A) by ENS_1:48;
A45:
EnsHom A = CatStr(#
(Hom A),
(Maps (Hom A)),
(fDom (Hom A)),
(fCod (Hom A)),
(fComp (Hom A)) #)
by ENS_1:def 13;
then A46:
cod f1 =
(fCod (Hom A)) . f1
.=
cod f19
by ENS_1:def 10
.=
(f1 `1) `2
by ENS_1:def 4
.=
[(Hom ((cod f),b)),(Hom ((dom f),b))] `2
by A44
.=
Hom (
(dom f),
b)
;
A47:
dom f4 =
(fDom (Hom A)) . f4
by A45
.=
dom f49
by ENS_1:def 9
.=
(f4 `1) `1
by ENS_1:def 3
.=
[(Hom ((cod f),a)),(Hom ((dom f),a))] `1
by A42
.=
Hom (
(cod f),
a)
;
A48:
cod f4 =
(fCod (Hom A)) . f4
by A45
.=
cod f49
by ENS_1:def 10
.=
(f4 `1) `2
by ENS_1:def 4
.=
[(Hom ((cod f),a)),(Hom ((dom f),a))] `2
by A42
.=
Hom (
(dom f),
a)
;
reconsider f2 =
<|(cod f),?> /. g as
Morphism of
(EnsHom A) ;
A49:
f2 =
(hom?- (cod f)) . g
by A8, CAT_3:def 10
.=
[[(Hom ((cod f),(dom g))),(Hom ((cod f),(cod g)))],(hom ((cod f),g))]
by ENS_1:def 21
;
then reconsider f29 =
f2 as
Element of
Maps (Hom A) by ENS_1:47;
A50:
dom f2 =
(fDom (Hom A)) . f2
by A45
.=
dom f29
by ENS_1:def 9
.=
(f2 `1) `1
by ENS_1:def 3
.=
[(Hom ((cod f),(dom g))),(Hom ((cod f),(cod g)))] `1
by A49
.=
Hom (
(cod f),
(dom g))
;
A51:
cod f2 =
(fCod (Hom A)) . f2
by A45
.=
cod f29
by ENS_1:def 10
.=
(f2 `1) `2
by ENS_1:def 4
.=
[(Hom ((cod f),(dom g))),(Hom ((cod f),(cod g)))] `2
by A49
.=
Hom (
(cod f),
(cod g))
;
A52:
dom f1 =
(fDom (Hom A)) . f1
by A45
.=
dom f19
by ENS_1:def 9
.=
(f1 `1) `1
by ENS_1:def 3
.=
[(Hom ((cod f),b)),(Hom ((dom f),b))] `1
by A44
.=
Hom (
(cod f),
b)
;
then A53:
cod f2 = dom f1
by A8, A51, CAT_1:5;
reconsider f3 =
<|(dom f),?> /. g as
Morphism of
(EnsHom A) ;
A54:
f3 =
(hom?- (dom f)) . g
by A8, CAT_3:def 10
.=
[[(Hom ((dom f),(dom g))),(Hom ((dom f),(cod g)))],(hom ((dom f),g))]
by ENS_1:def 21
;
then reconsider f39 =
f3 as
Element of
Maps (Hom A) by ENS_1:47;
A55:
cod f3 =
(fCod (Hom A)) . f3
by A45
.=
cod f39
by ENS_1:def 10
.=
(f3 `1) `2
by ENS_1:def 4
.=
[(Hom ((dom f),(dom g))),(Hom ((dom f),(cod g)))] `2
by A54
.=
Hom (
(dom f),
(cod g))
;
A56:
dom f3 =
(fDom (Hom A)) . f3
by A45
.=
dom f39
by ENS_1:def 9
.=
(f3 `1) `1
by ENS_1:def 3
.=
[(Hom ((dom f),(dom g))),(Hom ((dom f),(cod g)))] `1
by A54
.=
Hom (
(dom f),
(dom g))
;
then A57:
cod f4 = dom f3
by A8, A48, CAT_1:5;
Hom (
(<|(cod f),?> . b),
(<|(dom f),?> . b))
<> {}
by A1;
then (t . b) * (<|(cod f),?> /. g) =
f1 (*) f2
by A9, CAT_1:def 13
.=
[[(Hom ((cod f),(dom g))),(Hom ((dom f),b))],((hom (f,b)) * (hom ((cod f),g)))]
by A44, A52, A46, A49, A50, A51, A53, Th1
.=
[[(Hom ((cod f),a)),(Hom ((dom f),(cod g)))],((hom ((dom f),g)) * (hom (f,a)))]
by A10, A41, A21, A26, FUNCT_1:2
.=
f3 (*) f4
by A54, A56, A55, A42, A47, A48, A57, Th1
.=
(<|(dom f),?> /. g) * (t . a)
by A40, A43, CAT_1:def 13
;
hence
(t . b) * (<|(cod f),?> /. g) = (<|(dom f),?> /. g) * (t . a)
;
verum
end;
hence
<|(cod f),?> is_naturally_transformable_to <|(dom f),?>
by A7, NATTRA_1:def 7; verum