Disclaimer: I know practically nothing about category theory, so sorry if the answer is "obvious".

Consider a category C, a subcategory D and a "closure" (i.e. idempotent) functor F : C → D. Then, given an object X ∈ D, what is the "smallest" Y ∈ C such that F(Y) = X?

It's not clear that "smallest" is well-defined for all such categories and idempotent functors, but here are two motivational examples from algebra where the categories are concrete and the functors are a form of set extension:

C the category of fields, D the category of algebraically closed fields, F algebraic closure.

C the category of (integral) domains, D the category of fields, F "field of fractions".

There are more examples of the sort of "closure functors" I'm talking about in the answers to this question.