In the Wikipedia articles on direct and inverse limits there is a set representation of such limits for particular categories of algebraic objects. That is, the inverse limit is given as a subset of the cartesian product (when forgetting the algebraic structure) and the direct limit is given as a quotient of the disjoint union (resp. direct sum).
The same representation can also be given for the category of topological spaces (with morphisms being continuous maps) and measurable spaces (with morphisms being measurable maps).
I would like to know in what type of categories such a representation is valid? In particular, is it true that this representation always holds for all concrete categories when the cartesian product is taken to be the categorical product and the disjoint union the coproduct? Or does such a category has to satisfy some additional property (in order to be "relatable" to the category of sets (with products, coproducts and equivalence relations))? If a category does not satisfy such a property, can we still give a nice explicit characterization of such direct and inverse limits in terms of subsets of sets (with additional structure carried over)?