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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.

Johnny T.
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    Once you have that, then $\psi\circ \phi=\operatorname{Id}_D$ by the uniqueness part of the universal property. – jgon Jul 05 '20 at 14:11
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    To elaborate on the previous comment, have you never proved that any object satisfying a universal property is unique up to unique isomorphism? Or is there an additional subtlety here that I'm not catching? – Tabes Bridges Jul 05 '20 at 18:27

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