Non-trivial non-bijective isomorphisms in a category

I'm currently looking at the differences between the treatment of the notion of isomorphism in category theory and set theory. One interesting difference is how the two systems treat isomorphism. An important class of differences are where bijective homomorphisms are not always isomorphisms (e.g. Top), however, I'm interested in the case where we have a (category-theoretic) isomorphism that is not bijective. The two examples I've managed to come up with so far are:

(1.) The incredibly trivial: e.g. The category consisting of a single set $\{0,1\}$, whose only morphism is defined by $f(0)=0$ and $f(1) = 0$.

(2.) The not-really-involving-functions-at-all-trivial: e.g. The category of Proofs, where objects are sentences and arrows are equivalence classes of proofs. Isomorphisms here are equivalence classes of proofs between equivalent propositions, so aren't bijections, but only because they aren't really functions.

Are there interesting examples of non-bijective isomorphisms from elsewhere in category theory (where the example has interesting mathematical content, but this isn't just because the arrows do not have underlying set-theoretic functions)?

• Functors preserve isos, so of constructs (concrete categories with underlying functor to category $\mathbf{Sets}$) nothing can be expected. All isos will be sent to isos in $\mathbf{Sets}$ which are the bijections. – drhab Sep 27 '17 at 8:11
• I don't understand your first example. What category are you considering here ? – Arnaud D. Sep 27 '17 at 8:58
• @drhab Thanks: What about the case where there's no composition preserving functor back to set, but nonetheless an underlying set? Any interesting mathematics lurking there? – Neil Barton Sep 27 '17 at 9:46
• Might be, but I have never been on that territory (and also never encountered it) so sorry, but I can't help you. – drhab Sep 27 '17 at 9:51
• Arguably, it makes no sense to speak of "underlying sets" when you are not refering to a functor into $\mathsf{Set}$. You could map objects 'randomly' to any sets you like; I don't see what the point of this is. Morphisms between things (categories e.g.) should respect all the structure, not just some of it (only objects e.g.). - Isomorphisms may only be non-bijective simply because they don't have underlying functions. I don't think you can say anything more meaningful than that. – Stefan Perko Sep 27 '17 at 10:25