Suppose I have $A_1 \to B$ and $A_2 \to B$ in some category where fiber products exist. I am stuck on trying to prove that it is unique up to unique isomorphism. Suppose I have $C$ with $\pi_i: C \to A_i$ and $D$ with $p_i: D \to A_i$ both satisfying the universal property as a fiber product of $A_1 \to B$ and $A_2 \to B$. I want to show $C$ and $D$ are isomorphic.
Let $\phi: D \to C$ and $\psi: C \to D$ obtained by the universal property. What I can get by drawing commutative diagrams is $$ p_i \circ \psi \circ \phi = p_i \circ Id_D. $$
How can I prove $\psi \circ \phi = Id_D$ from here? Thank you.