Show that fiber products exist in the category of abelian groups.

Question

Show that fiber products exist in the category of abelian groups. In fact, If $X, Y$ are abelian groups with homomorphisms $f: X \to Z$ and $g: Y \to Z$ show that $X \times_z Y$ is the set of all pairs $(x, y)$ with $x \in X$ and $y \in Y$ such that $f(x) = g(y)$.

Answer 1

Hint: You definitively know that $\mathbf{Ab}$ has products and equalizers (kernels). Use this to show that it has fiber products by constructing the fiber product construction as the equalizer of some product of maps.