 Solution

The diagonal satisfies $pd=qd=1_Y$, so it equalizes the two projections. If $h:Z\to Y\times Y$ satisfies $ph=qh$, put $u=ph=qh$. The universal property of the product in a category gives $h=(u,u)=du$. Any other factor $v$ with $dv=h$ must satisfy $v=pdv=ph=u$. Thus the diagonal is the equalizer of the two product projections.