Pullback square with two identical sides

Question

I'm wondering if any known property holds for a pullback square of the following form: $$ \require{AMScd} \begin{CD} Y @>{g}>> X \\ @V{h}VV @V{f}VV \\ Y @>{g}>> X \end{CD} $$ In particular, when working in a topos with $g$ mono and $f$ and $h$ idempotent arrows, can we deduce something more? I was really hoping that $h$ would be an isomorphism, but I can't find neither a proof nor a counterexample

Answer 1

Here's a simple source of counterexamples.

Let $Y = X$ and $g = 1_X$. Then you have a pullback square if and only if $h=f$. (and also, the square commutes if and only if $h=f$)