I recently found myself at a spot that I never believed I'll get (or at least not that soon in my career). I ran into a problem which seems to be best answered via categories.
The situation is this, I have a directed system of structures and the maps are all the inclusion map, that is $X_i$ for $i\in I$ where $(I,\leq)$ is a directed set; and if $i\leq j$ then $X_i$ is a substructure of $X_j$.
Suppose that the direct limit of the system exists. Can I be certain that this direct limit is actually the union? Namely, what sort of categories would ensure this, and what possible counterexamples are there?
I asked several folks around the department today, some assured me that this is the case for concrete categories, while others assured me that a counterexample can be found (although it won't be organic, and would probably be manufactured for this case).
The situation is such that the direct system is originating from forcing, so it's quite... wide and probably immune to some of the "thinkable" counterexamples (by arguments of [set theoretical] genericity from one angle or another), and so any counterexample which is essentially a linearly ordered system is not going to be useful as a counterexample.
Another typical counterexample which is irrelevant here is finitely-generated things, for example we can take a direct system of f.g. vector spaces whose limit is not f.g. but this aspect is also irrelevant to me; although I am not sure how to accurately describe this sort of irrelevance.
Last point (which came up with every person I discussed this question today), if we consider: $\mathbb R\hookrightarrow\mathbb R^2\hookrightarrow\ldots$ Then we consider those to be actually increasing sets in inclusion and not "natural identifications" as commonly done in categories. So the limit of the above would actually be $\mathbb R^{<\omega}$ (all finite sequences from $\mathbb R$).