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Can skeleta simplify category theory?

When one first learns about categories, one confusing fact is that the nicest notion of two categories being the same is not isomorphism - it is rather equivalence. But it's occurred to me that when two categories are equivalent but not isomorphic, it seems that the only reason why this is so is because one category has multiple copies of the same object.

Could we make a notion of some type of category in which no two different objects are isomorphic? Let's call such a category *succinct*, since I don't know any other term for it (I hope succinct doesn't have another meaning). Then would it be true that any two succinct categories which are equivalent are also isomorphic? And if we have any category, we can form a new succinct category by identifying isomorphic objects. Then the question arises, why don't we just always work with succinct categories? Why, if we're interested in the category of sets, don't we say that any two sets with the same cardinality are not only isomorphic, but are actually the *same* object in the category of sets?