Suppose that $\mathcal{C}$ is a balanced category with epimorphic images, that is, every bimorphism is an isomorphism, every morphism has an image and the image factorization is an epi-mono factorization. Under such hypotheses, is it true that epimorphisms are stable under pullback (if it exists)? Or at least that $ f(f^{-1}(S)) = S $, if it's defined, where $ f : A \rightarrow B $ is an epimorphism and $S$ is a subobject of $B$? I believe both should be false, but I was not able to provide a counterexample.


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