It's well-known that, for lots of concrete categories (but by no means all), we can think of the objects as themselves being small categories, and morphisms are the functors between these categories. Examples include Grp, Ab, Top... When we apply such a construction, we turn a 1-category into a (strict, I think?) 2-category. But 2-categories carry some extra structure, namely the notion of a natural transformation. When we "decategorify" back down, where does this extra structure go?

I can work it out in some specific cases; for instance, if we categorify Ab in the obvious way, there are no nontrivial natural transformations. I don't have a characterization for when two morphisms of groups are naturally isomorphic as functors between the underlying categories, although I have a feel for how the question behaves.

Are there any sort of general results on what natural transformations between morphisms look like if we categorify thusly? Is it at least independent of how we realize the objects as small categories? (I suspect the answer to the second question is "no," but don't have the skills to construct a counterexample. I hope I'm wrong, though.)

More generally, if we can categorify an n-category into an (n+k)-category forgetting the higher morphisms, do the higher morphisms go downstairs to the n-category in any nice way?

It's well-known that, for lots of concrete categories (but by no means all), we can think of the objects as themselves being small categories, and morphisms are the functors between these categories. Examples include Grp, Ab, Top... When we apply such a construction, we turn a 1-category into a (strict, I think?) 2-category. But 2-categories carry some extra structure, namely the notion of a natural transformation. When we "decategorify" back down, where does this extra structure go?

I can work it out in some specific cases; for instance, if we categorify Ab in the obvious way, there are no nontrivial natural transformations. I don't have a characterization for when two morphisms of groups are naturally isomorphic as functors between the underlying categories, although I have a feel for how the question behaves.

Are there any sort of general results on what natural transformations between morphisms look like if we categorify thusly? Is it at least independent of how we realize the objects as small categories? (I suspect the answer to the second question is "no," but don't have the skills to construct a counterexample. I hope I'm wrong, though.)

More generally, if we can categorify an n-category into an (n+k)-category forgetting the higher morphisms, do the higher morphisms go downstairs to the n-category in any nice way?