**Note**: As pointed out in a comment by Peter and echoed by Andrew in his answer, the question as stated does not make sense because "being bimorphic to" is not an equivalence relation. Nevertheless, I will leave the question as stated in case it continues to receive interesting answers.

### Justification

This coming academic year I will be supervising a fourth-year undergraduate project on Category Theory. As should be clear to MO regulars, I'm far from an expert in this topic, but I am keen to improve my fluency in this language: hence the project.

Maybe it's only me, but I find that the unifying beauty of category theory can best be appreciated when one has lots of concrete (not necessarily in the categorical sense!) examples. Most books I've seen take this view as well and present a number of examples taken from algebra and topology, mostly. Given the origins of the subject, this is of course not surprising. From time to time, though, I find myself wanting more "realistic" examples for certain concepts. This question is about one such concept.

### Background

Recall that a *bimorphism* in a category is a morphism which is both monic (i.e., left cancellable) and epic (i.e., right cancellable). A closely related notion to (and, in a concrete category, a special case of) bimorphism is that of isomorphism: a morphism $f:A \to B$ is an isomorphism if there exists a morphism $f^{-1}: B \to A$ such that $f \circ f^{-1} = 1_B$ and $f^{-1} \circ f = 1_A$, with $1_{A,B}$ the corresponding identity morphisms. Categories in which all bimorphisms are isomorphisms are said to be *balanced*.

We say that two objects $A,B$ in a category are isomorphic if $\operatorname{Mor}(A,B)$ contains an isomorphism. Now, if the category is not balanced, it could very well be that $\operatorname{Mor}(A,B)$ contains a bimorphism even if it does not contain an isomorphism. For lack of a better name, let us call such objects *bimorphic* and let us speak of bimorphism classes of objects,...

Needless to say, the concept of isomorphism classes is very important and sits at the centre of any classification problem, but what about the concept of bimorphism classes?

### Question

Is there a natural context in which one is compelled to relax the notion of isomorphism to that of bimorphism? In particular, any geometric context?

Thanks in advance!