Are isomorphisms stable under pullbacks?
Actually, an isomorphism always has a pull-back (no assumption needed on the category) : if $f: X\to Y$ is an isomorphism, and $g:S\to Y$ is any morphism, then $$\require{AMScd} \begin{CD} S @>{f^{-1}\circ g}>> X\\ @V{Id}VV @V{f}VV \\ S @>{g}>> Y \end{CD}$$ is clearly a pull-back diagram (the universal property is easy to check).
So by unicity of fiber product, any pull-back of $f$ along $g$ must be an isomorphism.