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Let $f:A\to B$ be a morphism in a category $\mathcal{C}$ . The kernel pair of $f$ is defined as the pair of morphisms $(k_1: K\to A, k_2:K\to A)$ such that
is a pullback diagram.
Since
is a commutative diagram, we have a unique morphism $g:A\to K$ such that
is commutative. As a result, $k_1$ and $k_2$ are both monomorphisms: if $k_1\circ h_1 = k_1\circ h_2$ , then $$h_1 = 1_A \circ h_1 = (g\circ k_1) \circ h_1 = g\circ (k_1 \circ h_1) =g\circ (k_1 \circ h_2) = (g\circ k_1) \circ h_2 = 1_A \circ h_2 = h_2.$$
For example, in Set, the category of sets, the kernel pair of a function $f:A\to B$ is the pair $p_1:K\to A$ and $p_2:K\to A$ , given by $$K=\lbrace (a,b) \in A\times A \mid f(a)=f(b) \rbrace,$$ and $p_1$ and $p_2$ are given by $$p_1(a,b)=a \qquad \mbox{and} \qquad p_2(a,b)=b.$$ This is just the kernel of a function, in the sense of universal algebra. Please see this entry for more details.
The notion of cokernel pair is dually defined.
Remark. $f:A\to B$ is a monomorphism iff the kernel pair of $f$ is $(1_A,1_A)$ . Dually, $f$ is an epimorphism iff the cokernel pair of $f$ is $(1_A,1_A)$ .
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- F. Borceux Basic Category Theory, Handbook of Categorical Algebra I, Cambridge University Press, Cambridge (1994)
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