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2. Universal arrows for the
coproducts f
=
(in
1
, in
2
)
:
(A,B)
→
(A
+
B,A
+
B)
in
C
B
,
in
2
:
B
→
A
+
B
are two monomorphisms in
D
corresponding to the first and second
injections, respectively. Thus we obtain the following case for the commutative
diagram above in
C
=
D
×
D
, and a given object
X
=
(C,C)
∈
C
, where
in
1
:
A
→
A
+
=
D
×
D
(and its corresponding commutative diagram in
D
on the right), where
Y
2
=
(A,B)
and
Y
1
=
(A
+
B,A
+
B)
,
g
=
(
[
k,l
]
,
[
k,l
]
)
:
(A
+
B,A
+
B)
→
(C,C)
, with
k
:
A
→
C,l
:
B
→
C
two arrows in
D
:
The bijective function
h
X
(f
OP
)
:
C
(Y
1
,X)
→
C
(Y
2
,X)
in this case means that
for any arrow
(k,l)
:
(A,B)
→
(C,C)
in
C
(Y
2
,X)
(i.e., the couple of arrows
k
:
A
→
C
,
l
:
B
→
C
in
D
) there is the unique arrow
(
[
k,l
]
,
[
k,l
]
)
:
(A
+
B,A
+
B)
→
(C,C)
in
C
(Y
1
,X)
(i.e., the unique arrow
[
k,l
]:
A
+
B
→
C
in
D
), such
that the diagrams above commute.
For example, in
Set
the object
A
×
B
is the Cartesian product of the set
A
and set
B
, while
A
B
is the disjoint union of the set
A
and set
B
.
Products and coproducts are the particular cases of adjunctions. Given two func-
tors
F
+
C
, we say that
F
is
left adjoint
to
G
,or
G
is
right adjoint
to
F
, is there are the natural isomorphism:
θ
:
C
→
D
,
G
:
D
→
D
(F(
_
),
_
)
, where
:
C
(
_
,G(
_
))
C
OP
C
(
_
,G(
_
))
:
×
D
→
Set
is the result of composing the bivariant hom-functor
C
(
_
,
_
)
with
Id
C
OP
×
G
, and
D
(F(
_
),
_
)
is similar.
Equivalently, they are adjoint if there are two natural transformations,
counit ε
:
Id
D
and
unit η
GF
, such that the following diagrams of
FG
:
Id
C
natural transformations commute:
where
ηG
denotes the natural transformation with components (functions)
η
G(X)
:
G(X)
→
GFG(X)
and
G
◦
ε
denotes the natural transformation with components
G(ε
X
)
G(X)
, for each object
X
in
D
.
This adjunction is denoted by tuple
(F,G,ε,η)
, with
η
C
:
:
GFG(X)
→
G(F(C))
a
universal arrow for each object
C
. In fact, for any object
D
in
D
and morphism
f
C
→
G(D
)
there exists a unique morphism
f
D
, where
D
:
C
→
:
D
→
=
F(C)
,
such that the following two adjoint diagrams commute: