If the category in question is concrete with underlying set functor $C \to Set$, and bimorphisms are given by one-to-one and onto functions, then the more interesting question is what happens inside a bimorphism class. Namely, what sort of $C$-objects are there for a given underlying set? In other words, what do the fibres of $C\to Set$ look like? This is definitely an interesting question for including in a project.
As Peter points out in his comment, a space is bimorphic to the indiscrete space on the same set of points. But it is also bimorphic to the discrete space on the same set of points, and these form the top and bottom respectively of the poset of topologies on a set. This poset is very interesting, and has interesting topologies. This sort of behaviour would crop up with probably any topological concrete category over $Set$ (beware that these are often called 'topological categories' in the literature, but are neither categories internal to $Top$ nor $Top$-enriched categories)
Edit: I realised last night that we are not looking at the fibres of the underlying set functor, but the preimage of a whole isomorphism class of sets. This makes things a little bit more complicated, but also raises the interesting question of when are (essentially) endomorphisms of a set continuous with respect to two given topologies, or rather, what is the structure on the category consisting of spaces with the same underlying set and all maps between these covering endomorphisms of the underlying set? The poset of topologies is in there as those maps that cover the identity map.