When is a bijective homomorphism an isomorphism?

Recall that structures and their structure preserving maps often assemble themselves into categories. So, there is a category $Grp$ of groups, $Ab$ of abelian groups, $Ring$ of rings and so on. Now, if the structures are based on sets, then often there will be a forgetful functor $C\to Set$ from the category $C$ to the category of sets and functions. The property you are looking at is reflection of isomorphisms by this functor. So, one can ask, for a given category $C$, when is the forgetful functor $C\to Set$, assuming it exists, reflects isomorphism? A pretty far reaching answer is that whenever $C\to Set$ is monadic. Now, that latter term is a bit more technical, but, in a nutshell, $C\to Set$ is monadic if $C$ is a category of nice enough algebraic structures. Monadicity captures many algebraic structures, but not, for instance, posets (if you consider these algebraic): the forgetful functor does not reflect isomorphisms.

Interestingly, this notion of reflection of isomorphisms is in fact one of the conditions of Beck's Monadicity Theorem characterising monadic adjunctions.

A relevant anecdote but not an answer.

When I took abstract algebra (in 1956, way before categories) a question on the first exam asked for the definition of an isomorphism for an equivalence relation - not something we'd covered in class.

I (and most of my classmates) naively modified the definition we knew for group homomorphisms, requiring that the bijection $\phi$ satisfy $\phi(x) \equiv \phi(y)$ whenever $x \equiv y$. We all lost points.