Actually, the analogy to deformation retracts in Reid's answer can be made quite precise. In the canonical or categorical model structure on Cat (often problematically called the "folk model structure"), the acyclic cofibrations are the equivalences of categories which are injective on objects. These are evidently precisely the functors that occur as F in your equimorphisms.
So probably a more correct thing to say than my original answer would be that it depends on whether you are treating categories truly "categorically," or whether you are using "too-strict" notions for convenience in getting at the weaker "correct" notions. Model category theory, and homotopy theory more generally, is all about doing the latter. Category theorists tend to work directly with the "right" notions, because for ordinary categories doing so is pretty easy--but of course when you get up to higher categories, some model category theory frequently turns out to be useful.
Your original question, though, suggested that you were thinking about whether such equimorphisms "arise naturally" between the ordinary sort of large categories that appear in mathematical practice. I think I would still say that whenever that happens, it's an accident of the chosen set-theory rather than anything really interesting. The "too-strict" notions really only have technical, rather than fundamental, importance.
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