Stacks *qua* moduli spaces were introduced to keep track of nontrivial automorphisms of the objects they parameterize. In essence they are groupoids of objects with some form geometric cohesion. The classic example is principal bundles/torsors, the whole category of which is actually a groupoid. But what about objects one might want to parameterise which have non-invertible maps between them, such as vector bundles, or coherent sheaves? One could imagine a stack of such objects, because they 'glue' as principal bundles do, but if one keeps track of all maps this thing should be a category, not a groupoid. It is certainly deserving of being promoted to something geometric, and so one could present it by a category in algebraic spaces or schemes, much as the stacks we are more familiar with are presented by algebraic groupoids.

Some might argue that we have classifying topoi or similar for these situations, and this is well and good, but what about some geometry on these topoi? I know of two different takes on classifying topoi of a small (internal) category, one approach involving flat functors and the other torsors for the groupoid of all invertible arrows of the category in question. Between these two competing definitions, there are arguments (at least in my own head, too fuzzy to unveil here) both ways, and concrete examples of where one uses stacks of categories in a geometric context would certainly push the balance in one direction or another. This isn't the only reason I would like to know an answer to this question, but it has some bearing - and I've rabbited on long enough.

Question: Do stacks $\operatorname{Sch}^{op} \to \operatorname{Cat}$ of categories come up anywhere in algebraic geometry, such that one considers presentations by internal categories in $\operatorname{Sch}$? If yes, how is the presentation specified? If no, why not? And is there anything stopping us from doing so? (Technical reasons, terminological, or we have other techniques that are better and so on)