Characterize cokernel pair in Set

Question

In the category of sets (Set), can we characterize the cokernel pair of a function $f : A\rightarrow B$?

Answer 1

Let $s, t:B\rightarrow C$ be the cokernel pair.

$C$ is the coproduct $B + B$, but with the image of $f$ identified (glued together), in other words:

$$C = \mathrm{im} f + (B - \mathrm{im} f) + (B - \mathrm{im} f)$$

The two morphism $s, t$ are simply the left and right coprojections into $C$.

To see why this is, observe that the cokernel pair is defined as the pushout of the function along itself:

$$ \array{ &&&& A &&&& \\ & && f \swarrow & & \searrow f && \\ && B &&&& B \\ & && s\searrow & & \swarrow t && \\ &&&& C &&&& } $$

We know that in Set, $C$ is defined as the set $B + B$ up to the equivalence $b_1 \cong b_2$ iff there exists $a \in A$ such that $f(a) = b_1$ and $f(a) = b_2$; in other words, iff $b_1 = b_2 \in \mathrm{im} f$.

This means that $C$ has one copy of $\mathrm{im} f$ and two copies of every other element.