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?
Best Answer
Hint
Let's call $h: E \rightarrow B$. Now considering $B \times B$ from the definition know the existence of two maps $\pi_1, \pi_2 : B \times B \rightarrow B$ such that:
$\pi_1 \circ <f,g> = f$
$\pi_2 \circ <f,g> = g$
Knowing this we can see:
$f \circ e = \pi_1 \circ <f,g> \circ e = \pi_1 \circ \Delta \circ h = h= \pi_2 \circ \Delta \circ h = \pi_2 \circ <f,g> \circ e = g \circ e$