== Exercises ==
Given
f
:
a
→
b
{\displaystyle f:a\to b}
in an
A
b
{\displaystyle Ab}
-enriched category with zero object. Prove that
f
=
0
a
,
b
{\displaystyle f=0_{a,b}}
iff
f
{\displaystyle f}
factors through
0
{\displaystyle \mathbf {0} }
.Given a biproduct
(
a
⊕
b
,
i
1
,
p
1
,
i
2
,
p
2
)
{\displaystyle (a\oplus b,i_{1},p_{1},i_{2},p_{2})}
of
a
{\displaystyle a}
and
b
{\displaystyle b}
. Prove that
(
a
⊕
b
,
i
1
,
i
2
)
{\displaystyle (a\oplus b,i_{1},i_{2})}
is a coproduct of
a
{\displaystyle a}
and
b
{\displaystyle b}
and
(
a
⊕
b
,
p
1
,
p
2
)
{\displaystyle (a\oplus b,p_{1},p_{2})}
is a product of
a
{\displaystyle a}
and
b
{\displaystyle b}
.In an
A
b
{\displaystyle Ab}
-enriched category with zero object, a kernel of
f
:
a
→
b
{\displaystyle f:a\to b}
can be equivalently be characterized as a pullback of
0
→
b
{\displaystyle \mathbf {0} \to b}
along
f
{\displaystyle f}
.