:: CAT_5 semantic presentation
theorem Th1: :: CAT_5:1
for b
1, b
2 being
CatStr holds
(
CatStr(# the
Objects of b
1,the
Morphisms of b
1,the
Dom of b
1,the
Cod of b
1,the
Comp of b
1,the
Id of b
1 #)
= CatStr(# the
Objects of b
2,the
Morphisms of b
2,the
Dom of b
2,the
Cod of b
2,the
Comp of b
2,the
Id of b
2 #) & b
1 is
Category-like implies b
2 is
Category-like )
:: deftheorem Def1 defines with_triple-like_morphisms CAT_5:def 1 :
theorem Th2: :: CAT_5:2
scheme :: CAT_5:sch 1
s1{ F
1()
-> non
empty set , F
2()
-> non
empty set , P
1[
set ,
set ,
set ], F
3(
set ,
set )
-> set } :
ex b
1 being
strict with_triple-like_morphisms Category st
( the
Objects of b
1 = F
1() & ( for b
2, b
3 being
Element of F
1()
for b
4 being
Element of F
2() holds
( P
1[b
2,b
3,b
4] implies
[[b2,b3],b4] is
Morphism of b
1 ) ) & ( for b
2 being
Morphism of b
1 holds
ex b
3, b
4 being
Element of F
1()ex b
5 being
Element of F
2() st
( b
2 = [[b3,b4],b5] & P
1[b
3,b
4,b
5] ) ) & ( for b
2, b
3 being
Morphism of b
1for b
4, b
5, b
6 being
Element of F
1()
for b
7, b
8 being
Element of F
2() holds
( b
2 = [[b4,b5],b7] & b
3 = [[b5,b6],b8] implies b
3 * b
2 = [[b4,b6],F3(b8,b7)] ) ) )
provided
E4:
for b
1, b
2, b
3 being
Element of F
1()
for b
4, b
5 being
Element of F
2() holds
( P
1[b
1,b
2,b
4] & P
1[b
2,b
3,b
5] implies ( F
3(b
5,b
4)
in F
2() & P
1[b
1,b
3,F
3(b
5,b
4)] ) )
and
E5:
for b
1 being
Element of F
1() holds
ex b
2 being
Element of F
2() st
( P
1[b
1,b
1,b
2] & ( for b
3 being
Element of F
1()
for b
4 being
Element of F
2() holds
( ( P
1[b
1,b
3,b
4] implies F
3(b
4,b
2)
= b
4 ) & ( P
1[b
3,b
1,b
4] implies F
3(b
2,b
4)
= b
4 ) ) ) )
and
E6:
for b
1, b
2, b
3, b
4 being
Element of F
1()
for b
5, b
6, b
7 being
Element of F
2() holds
( P
1[b
1,b
2,b
5] & P
1[b
2,b
3,b
6] & P
1[b
3,b
4,b
7] implies F
3(b
7,F
3(b
6,b
5))
= F
3(F
3(b
7,b
6),b
5) )
scheme :: CAT_5:sch 2
s2{ F
1()
-> non
empty set , F
2()
-> non
empty set , P
1[
set ,
set ,
set ], F
3(
set ,
set )
-> set } :
for b
1, b
2 being
strict with_triple-like_morphisms Category holds
( the
Objects of b
1 = F
1() & ( for b
3, b
4 being
Element of F
1()
for b
5 being
Element of F
2() holds
( P
1[b
3,b
4,b
5] implies
[[b3,b4],b5] is
Morphism of b
1 ) ) & ( for b
3 being
Morphism of b
1 holds
ex b
4, b
5 being
Element of F
1()ex b
6 being
Element of F
2() st
( b
3 = [[b4,b5],b6] & P
1[b
4,b
5,b
6] ) ) & ( for b
3, b
4 being
Morphism of b
1for b
5, b
6, b
7 being
Element of F
1()
for b
8, b
9 being
Element of F
2() holds
( b
3 = [[b5,b6],b8] & b
4 = [[b6,b7],b9] implies b
4 * b
3 = [[b5,b7],F3(b9,b8)] ) ) & the
Objects of b
2 = F
1() & ( for b
3, b
4 being
Element of F
1()
for b
5 being
Element of F
2() holds
( P
1[b
3,b
4,b
5] implies
[[b3,b4],b5] is
Morphism of b
2 ) ) & ( for b
3 being
Morphism of b
2 holds
ex b
4, b
5 being
Element of F
1()ex b
6 being
Element of F
2() st
( b
3 = [[b4,b5],b6] & P
1[b
4,b
5,b
6] ) ) & ( for b
3, b
4 being
Morphism of b
2for b
5, b
6, b
7 being
Element of F
1()
for b
8, b
9 being
Element of F
2() holds
( b
3 = [[b5,b6],b8] & b
4 = [[b6,b7],b9] implies b
4 * b
3 = [[b5,b7],F3(b9,b8)] ) ) implies b
1 = b
2 )
provided
E4:
for b
1 being
Element of F
1() holds
ex b
2 being
Element of F
2() st
( P
1[b
1,b
1,b
2] & ( for b
3 being
Element of F
1()
for b
4 being
Element of F
2() holds
( ( P
1[b
1,b
3,b
4] implies F
3(b
4,b
2)
= b
4 ) & ( P
1[b
3,b
1,b
4] implies F
3(b
2,b
4)
= b
4 ) ) ) )
theorem Th3: :: CAT_5:3
for b
1 being
Categoryfor b
2 being
Subcategory of b
1 holds
( b
1 is
Subcategory of b
2 implies
CatStr(# the
Objects of b
1,the
Morphisms of b
1,the
Dom of b
1,the
Cod of b
1,the
Comp of b
1,the
Id of b
1 #)
= CatStr(# the
Objects of b
2,the
Morphisms of b
2,the
Dom of b
2,the
Cod of b
2,the
Comp of b
2,the
Id of b
2 #) )
theorem Th4: :: CAT_5:4
:: deftheorem Def2 defines /\ CAT_5:def 2 :
theorem Th5: :: CAT_5:5
theorem Th6: :: CAT_5:6
:: deftheorem Def3 defines Image CAT_5:def 3 :
theorem Th7: :: CAT_5:7
theorem Th8: :: CAT_5:8
theorem Th9: :: CAT_5:9
:: deftheorem Def4 defines categorial CAT_5:def 4 :
:: deftheorem Def5 defines categorial CAT_5:def 5 :
definition
let c
1 be
Category;
attr a
1 is
Categorial means :
Def6:
:: CAT_5:def 6
( the
Objects of a
1 is
categorial & ( for b
1 being
Object of a
1for b
2 being
Category holds
( b
1 = b
2 implies
id b
1 = [[b2,b2],(id b2)] ) ) & ( for b
1 being
Morphism of a
1for b
2, b
3 being
Category holds
not ( b
2 = dom b
1 & b
3 = cod b
1 & ( for b
4 being
Functor of b
2,b
3 holds
not b
1 = [[b2,b3],b4] ) ) ) & ( for b
1, b
2 being
Morphism of a
1for b
3, b
4, b
5 being
Categoryfor b
6 being
Functor of b
3,b
4for b
7 being
Functor of b
4,b
5 holds
( b
1 = [[b3,b4],b6] & b
2 = [[b4,b5],b7] implies b
2 * b
1 = [[b3,b5],(b7 * b6)] ) ) );
end;
:: deftheorem Def6 defines Categorial CAT_5:def 6 :
for b
1 being
Category holds
( b
1 is
Categorial iff ( the
Objects of b
1 is
categorial & ( for b
2 being
Object of b
1for b
3 being
Category holds
( b
2 = b
3 implies
id b
2 = [[b3,b3],(id b3)] ) ) & ( for b
2 being
Morphism of b
1for b
3, b
4 being
Category holds
not ( b
3 = dom b
2 & b
4 = cod b
2 & ( for b
5 being
Functor of b
3,b
4 holds
not b
2 = [[b3,b4],b5] ) ) ) & ( for b
2, b
3 being
Morphism of b
1for b
4, b
5, b
6 being
Categoryfor b
7 being
Functor of b
4,b
5for b
8 being
Functor of b
5,b
6 holds
( b
2 = [[b4,b5],b7] & b
3 = [[b5,b6],b8] implies b
3 * b
2 = [[b4,b6],(b8 * b7)] ) ) ) );
theorem Th10: :: CAT_5:10
for b
1, b
2 being
Category holds
(
CatStr(# the
Objects of b
1,the
Morphisms of b
1,the
Dom of b
1,the
Cod of b
1,the
Comp of b
1,the
Id of b
1 #)
= CatStr(# the
Objects of b
2,the
Morphisms of b
2,the
Dom of b
2,the
Cod of b
2,the
Comp of b
2,the
Id of b
2 #) & b
1 is
Categorial implies b
2 is
Categorial )
theorem Th11: :: CAT_5:11
theorem Th12: :: CAT_5:12
theorem Th13: :: CAT_5:13
theorem Th14: :: CAT_5:14
for b
1, b
2 being
Categorial Category holds
( the
Objects of b
1 = the
Objects of b
2 & the
Morphisms of b
1 = the
Morphisms of b
2 implies
CatStr(# the
Objects of b
1,the
Morphisms of b
1,the
Dom of b
1,the
Cod of b
1,the
Comp of b
1,the
Id of b
1 #)
= CatStr(# the
Objects of b
2,the
Morphisms of b
2,the
Dom of b
2,the
Cod of b
2,the
Comp of b
2,the
Id of b
2 #) )
theorem Th15: :: CAT_5:15
:: deftheorem Def7 defines cat CAT_5:def 7 :
theorem Th16: :: CAT_5:16
theorem Th17: :: CAT_5:17
theorem Th18: :: CAT_5:18
:: deftheorem Def8 defines full CAT_5:def 8 :
theorem Th19: :: CAT_5:19
for b
1, b
2 being
Categorial full Category holds
( the
Objects of b
1 = the
Objects of b
2 implies
CatStr(# the
Objects of b
1,the
Morphisms of b
1,the
Dom of b
1,the
Cod of b
1,the
Comp of b
1,the
Id of b
1 #)
= CatStr(# the
Objects of b
2,the
Morphisms of b
2,the
Dom of b
2,the
Cod of b
2,the
Comp of b
2,the
Id of b
2 #) )
theorem Th20: :: CAT_5:20
theorem Th21: :: CAT_5:21
theorem Th22: :: CAT_5:22
:: deftheorem Def9 defines Hom CAT_5:def 9 :
:: deftheorem Def10 defines Hom CAT_5:def 10 :
theorem Th23: :: CAT_5:23
theorem Th24: :: CAT_5:24
theorem Th25: :: CAT_5:25
theorem Th26: :: CAT_5:26
theorem Th27: :: CAT_5:27
theorem Th28: :: CAT_5:28
definition
let c
1 be
Category;
let c
2 be
Object of c
1;
set c
3 =
Hom c
2;
set c
4 = the
Morphisms of c
1;
defpred S
1[
Element of
Hom c
2,
Element of
Hom c
2,
Element of the
Morphisms of c
1] means (
dom a
2 = cod a
3 & a
1 = a
2 * a
3 );
deffunc H
1(
Morphism of c
1,
Morphism of c
1)
-> Element of the
Morphisms of c
1 = a
1 * a
2;
E32:
for b
1, b
2, b
3 being
Element of
Hom c
2for b
4, b
5 being
Element of the
Morphisms of c
1 holds
( S
1[b
1,b
2,b
4] & S
1[b
2,b
3,b
5] implies ( H
1(b
5,b
4)
in the
Morphisms of c
1 & S
1[b
1,b
3,H
1(b
5,b
4)] ) )
E33:
for b
1 being
Element of
Hom c
2 holds
ex b
2 being
Element of the
Morphisms of c
1 st
( S
1[b
1,b
1,b
2] & ( for b
3 being
Element of
Hom c
2for b
4 being
Element of the
Morphisms of c
1 holds
( ( S
1[b
1,b
3,b
4] implies H
1(b
4,b
2)
= b
4 ) & ( S
1[b
3,b
1,b
4] implies H
1(b
2,b
4)
= b
4 ) ) ) )
E34:
for b
1, b
2, b
3, b
4 being
Element of
Hom c
2for b
5, b
6, b
7 being
Element of the
Morphisms of c
1 holds
( S
1[b
1,b
2,b
5] & S
1[b
2,b
3,b
6] & S
1[b
3,b
4,b
7] implies H
1(b
7,H
1(b
6,b
5))
= H
1(H
1(b
7,b
6),b
5) )
func c
1 -SliceCat c
2 -> strict with_triple-like_morphisms Category means :
Def11:
:: CAT_5:def 11
( the
Objects of a
3 = Hom a
2 & ( for b
1, b
2 being
Element of
Hom a
2for b
3 being
Morphism of a
1 holds
(
dom b
2 = cod b
3 & b
1 = b
2 * b
3 implies
[[b1,b2],b3] is
Morphism of a
3 ) ) & ( for b
1 being
Morphism of a
3 holds
ex b
2, b
3 being
Element of
Hom a
2ex b
4 being
Morphism of a
1 st
( b
1 = [[b2,b3],b4] &
dom b
3 = cod b
4 & b
2 = b
3 * b
4 ) ) & ( for b
1, b
2 being
Morphism of a
3for b
3, b
4, b
5 being
Element of
Hom a
2for b
6, b
7 being
Morphism of a
1 holds
( b
1 = [[b3,b4],b6] & b
2 = [[b4,b5],b7] implies b
2 * b
1 = [[b3,b5],(b7 * b6)] ) ) );
existence
ex b1 being strict with_triple-like_morphisms Category st
( the Objects of b1 = Hom c2 & ( for b2, b3 being Element of Hom c2
for b4 being Morphism of c1 holds
( dom b3 = cod b4 & b2 = b3 * b4 implies [[b2,b3],b4] is Morphism of b1 ) ) & ( for b2 being Morphism of b1 holds
ex b3, b4 being Element of Hom c2ex b5 being Morphism of c1 st
( b2 = [[b3,b4],b5] & dom b4 = cod b5 & b3 = b4 * b5 ) ) & ( for b2, b3 being Morphism of b1
for b4, b5, b6 being Element of Hom c2
for b7, b8 being Morphism of c1 holds
( b2 = [[b4,b5],b7] & b3 = [[b5,b6],b8] implies b3 * b2 = [[b4,b6],(b8 * b7)] ) ) )
uniqueness
for b1, b2 being strict with_triple-like_morphisms Category holds
( the Objects of b1 = Hom c2 & ( for b3, b4 being Element of Hom c2
for b5 being Morphism of c1 holds
( dom b4 = cod b5 & b3 = b4 * b5 implies [[b3,b4],b5] is Morphism of b1 ) ) & ( for b3 being Morphism of b1 holds
ex b4, b5 being Element of Hom c2ex b6 being Morphism of c1 st
( b3 = [[b4,b5],b6] & dom b5 = cod b6 & b4 = b5 * b6 ) ) & ( for b3, b4 being Morphism of b1
for b5, b6, b7 being Element of Hom c2
for b8, b9 being Morphism of c1 holds
( b3 = [[b5,b6],b8] & b4 = [[b6,b7],b9] implies b4 * b3 = [[b5,b7],(b9 * b8)] ) ) & the Objects of b2 = Hom c2 & ( for b3, b4 being Element of Hom c2
for b5 being Morphism of c1 holds
( dom b4 = cod b5 & b3 = b4 * b5 implies [[b3,b4],b5] is Morphism of b2 ) ) & ( for b3 being Morphism of b2 holds
ex b4, b5 being Element of Hom c2ex b6 being Morphism of c1 st
( b3 = [[b4,b5],b6] & dom b5 = cod b6 & b4 = b5 * b6 ) ) & ( for b3, b4 being Morphism of b2
for b5, b6, b7 being Element of Hom c2
for b8, b9 being Morphism of c1 holds
( b3 = [[b5,b6],b8] & b4 = [[b6,b7],b9] implies b4 * b3 = [[b5,b7],(b9 * b8)] ) ) implies b1 = b2 )
set c
5 = c
2 Hom ;
defpred S
2[
Element of c
2 Hom ,
Element of c
2 Hom ,
Element of the
Morphisms of c
1] means (
dom a
3 = cod a
1 & a
2 = a
3 * a
1 );
E35:
for b
1, b
2, b
3 being
Element of c
2 Hom for b
4, b
5 being
Element of the
Morphisms of c
1 holds
( S
2[b
1,b
2,b
4] & S
2[b
2,b
3,b
5] implies ( H
1(b
5,b
4)
in the
Morphisms of c
1 & S
2[b
1,b
3,H
1(b
5,b
4)] ) )
E36:
for b
1 being
Element of c
2 Hom holds
ex b
2 being
Element of the
Morphisms of c
1 st
( S
2[b
1,b
1,b
2] & ( for b
3 being
Element of c
2 Hom for b
4 being
Element of the
Morphisms of c
1 holds
( ( S
2[b
1,b
3,b
4] implies H
1(b
4,b
2)
= b
4 ) & ( S
2[b
3,b
1,b
4] implies H
1(b
2,b
4)
= b
4 ) ) ) )
E37:
for b
1, b
2, b
3, b
4 being
Element of c
2 Hom for b
5, b
6, b
7 being
Element of the
Morphisms of c
1 holds
( S
2[b
1,b
2,b
5] & S
2[b
2,b
3,b
6] & S
2[b
3,b
4,b
7] implies H
1(b
7,H
1(b
6,b
5))
= H
1(H
1(b
7,b
6),b
5) )
func c
2 -SliceCat c
1 -> strict with_triple-like_morphisms Category means :
Def12:
:: CAT_5:def 12
( the
Objects of a
3 = a
2 Hom & ( for b
1, b
2 being
Element of a
2 Hom for b
3 being
Morphism of a
1 holds
(
dom b
3 = cod b
1 & b
3 * b
1 = b
2 implies
[[b1,b2],b3] is
Morphism of a
3 ) ) & ( for b
1 being
Morphism of a
3 holds
ex b
2, b
3 being
Element of a
2 Hom ex b
4 being
Morphism of a
1 st
( b
1 = [[b2,b3],b4] &
dom b
4 = cod b
2 & b
4 * b
2 = b
3 ) ) & ( for b
1, b
2 being
Morphism of a
3for b
3, b
4, b
5 being
Element of a
2 Hom for b
6, b
7 being
Morphism of a
1 holds
( b
1 = [[b3,b4],b6] & b
2 = [[b4,b5],b7] implies b
2 * b
1 = [[b3,b5],(b7 * b6)] ) ) );
existence
ex b1 being strict with_triple-like_morphisms Category st
( the Objects of b1 = c2 Hom & ( for b2, b3 being Element of c2 Hom
for b4 being Morphism of c1 holds
( dom b4 = cod b2 & b4 * b2 = b3 implies [[b2,b3],b4] is Morphism of b1 ) ) & ( for b2 being Morphism of b1 holds
ex b3, b4 being Element of c2 Hom ex b5 being Morphism of c1 st
( b2 = [[b3,b4],b5] & dom b5 = cod b3 & b5 * b3 = b4 ) ) & ( for b2, b3 being Morphism of b1
for b4, b5, b6 being Element of c2 Hom
for b7, b8 being Morphism of c1 holds
( b2 = [[b4,b5],b7] & b3 = [[b5,b6],b8] implies b3 * b2 = [[b4,b6],(b8 * b7)] ) ) )
uniqueness
for b1, b2 being strict with_triple-like_morphisms Category holds
( the Objects of b1 = c2 Hom & ( for b3, b4 being Element of c2 Hom
for b5 being Morphism of c1 holds
( dom b5 = cod b3 & b5 * b3 = b4 implies [[b3,b4],b5] is Morphism of b1 ) ) & ( for b3 being Morphism of b1 holds
ex b4, b5 being Element of c2 Hom ex b6 being Morphism of c1 st
( b3 = [[b4,b5],b6] & dom b6 = cod b4 & b6 * b4 = b5 ) ) & ( for b3, b4 being Morphism of b1
for b5, b6, b7 being Element of c2 Hom
for b8, b9 being Morphism of c1 holds
( b3 = [[b5,b6],b8] & b4 = [[b6,b7],b9] implies b4 * b3 = [[b5,b7],(b9 * b8)] ) ) & the Objects of b2 = c2 Hom & ( for b3, b4 being Element of c2 Hom
for b5 being Morphism of c1 holds
( dom b5 = cod b3 & b5 * b3 = b4 implies [[b3,b4],b5] is Morphism of b2 ) ) & ( for b3 being Morphism of b2 holds
ex b4, b5 being Element of c2 Hom ex b6 being Morphism of c1 st
( b3 = [[b4,b5],b6] & dom b6 = cod b4 & b6 * b4 = b5 ) ) & ( for b3, b4 being Morphism of b2
for b5, b6, b7 being Element of c2 Hom
for b8, b9 being Morphism of c1 holds
( b3 = [[b5,b6],b8] & b4 = [[b6,b7],b9] implies b4 * b3 = [[b5,b7],(b9 * b8)] ) ) implies b1 = b2 )
end;
:: deftheorem Def11 defines -SliceCat CAT_5:def 11 :
for b
1 being
Categoryfor b
2 being
Object of b
1for b
3 being
strict with_triple-like_morphisms Category holds
( b
3 = b
1 -SliceCat b
2 iff ( the
Objects of b
3 = Hom b
2 & ( for b
4, b
5 being
Element of
Hom b
2for b
6 being
Morphism of b
1 holds
(
dom b
5 = cod b
6 & b
4 = b
5 * b
6 implies
[[b4,b5],b6] is
Morphism of b
3 ) ) & ( for b
4 being
Morphism of b
3 holds
ex b
5, b
6 being
Element of
Hom b
2ex b
7 being
Morphism of b
1 st
( b
4 = [[b5,b6],b7] &
dom b
6 = cod b
7 & b
5 = b
6 * b
7 ) ) & ( for b
4, b
5 being
Morphism of b
3for b
6, b
7, b
8 being
Element of
Hom b
2for b
9, b
10 being
Morphism of b
1 holds
( b
4 = [[b6,b7],b9] & b
5 = [[b7,b8],b10] implies b
5 * b
4 = [[b6,b8],(b10 * b9)] ) ) ) );
:: deftheorem Def12 defines -SliceCat CAT_5:def 12 :
for b
1 being
Categoryfor b
2 being
Object of b
1for b
3 being
strict with_triple-like_morphisms Category holds
( b
3 = b
2 -SliceCat b
1 iff ( the
Objects of b
3 = b
2 Hom & ( for b
4, b
5 being
Element of b
2 Hom for b
6 being
Morphism of b
1 holds
(
dom b
6 = cod b
4 & b
6 * b
4 = b
5 implies
[[b4,b5],b6] is
Morphism of b
3 ) ) & ( for b
4 being
Morphism of b
3 holds
ex b
5, b
6 being
Element of b
2 Hom ex b
7 being
Morphism of b
1 st
( b
4 = [[b5,b6],b7] &
dom b
7 = cod b
5 & b
7 * b
5 = b
6 ) ) & ( for b
4, b
5 being
Morphism of b
3for b
6, b
7, b
8 being
Element of b
2 Hom for b
9, b
10 being
Morphism of b
1 holds
( b
4 = [[b6,b7],b9] & b
5 = [[b7,b8],b10] implies b
5 * b
4 = [[b6,b8],(b10 * b9)] ) ) ) );
theorem Th29: :: CAT_5:29
theorem Th30: :: CAT_5:30
theorem Th31: :: CAT_5:31
theorem Th32: :: CAT_5:32
definition
let c
1 be
Category;
let c
2 be
Morphism of c
1;
func SliceFunctor c
2 -> Functor of a
1 -SliceCat (dom a2),a
1 -SliceCat (cod a2) means :
Def13:
:: CAT_5:def 13
for b
1 being
Morphism of
(a1 -SliceCat (dom a2)) holds a
3 . b
1 = [[(a2 * (b1 `11 )),(a2 * (b1 `12 ))],(b1 `2 )];
existence
ex b1 being Functor of c1 -SliceCat (dom c2),c1 -SliceCat (cod c2) st
for b2 being Morphism of (c1 -SliceCat (dom c2)) holds b1 . b2 = [[(c2 * (b2 `11 )),(c2 * (b2 `12 ))],(b2 `2 )]
uniqueness
for b1, b2 being Functor of c1 -SliceCat (dom c2),c1 -SliceCat (cod c2) holds
( ( for b3 being Morphism of (c1 -SliceCat (dom c2)) holds b1 . b3 = [[(c2 * (b3 `11 )),(c2 * (b3 `12 ))],(b3 `2 )] ) & ( for b3 being Morphism of (c1 -SliceCat (dom c2)) holds b2 . b3 = [[(c2 * (b3 `11 )),(c2 * (b3 `12 ))],(b3 `2 )] ) implies b1 = b2 )
func SliceContraFunctor c
2 -> Functor of
(cod a2) -SliceCat a
1,
(dom a2) -SliceCat a
1 means :
Def14:
:: CAT_5:def 14
for b
1 being
Morphism of
((cod a2) -SliceCat a1) holds a
3 . b
1 = [[((b1 `11 ) * a2),((b1 `12 ) * a2)],(b1 `2 )];
existence
ex b1 being Functor of (cod c2) -SliceCat c1,(dom c2) -SliceCat c1 st
for b2 being Morphism of ((cod c2) -SliceCat c1) holds b1 . b2 = [[((b2 `11 ) * c2),((b2 `12 ) * c2)],(b2 `2 )]
uniqueness
for b1, b2 being Functor of (cod c2) -SliceCat c1,(dom c2) -SliceCat c1 holds
( ( for b3 being Morphism of ((cod c2) -SliceCat c1) holds b1 . b3 = [[((b3 `11 ) * c2),((b3 `12 ) * c2)],(b3 `2 )] ) & ( for b3 being Morphism of ((cod c2) -SliceCat c1) holds b2 . b3 = [[((b3 `11 ) * c2),((b3 `12 ) * c2)],(b3 `2 )] ) implies b1 = b2 )
end;
:: deftheorem Def13 defines SliceFunctor CAT_5:def 13 :
:: deftheorem Def14 defines SliceContraFunctor CAT_5:def 14 :
theorem Th33: :: CAT_5:33
theorem Th34: :: CAT_5:34