I'm trying to solve exercise 5.6 in Steve Awodey's "Category Theory":
Show that a category with pull-backs and products has equalizers as follows: given arrows $f, g: A \to B$, take the pullback indicated below, where $\Delta = \langle 1_B, 1_B \rangle$:
Show that $e: E \to A$ is the equalizer of $f$ and $g$.
So I need to prove that $f \circ e = g \circ e$, and that given any $z: Z \to A$ with $f \circ z = g \circ z$ we can define a unique arrow such that $e \circ u = z$. How do I use the pullback property to do this?