Set representations of direct and inverse limits

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)?

• Your description of colimits in categories of algebraic objects is incorrect. Consider, for instance, coproducts in the category of abelian groups. – Zhen Lin Oct 23 '15 at 12:01
• I just copied the word "disjoint union" as was written in Wikipedia and added "direct sum" in parantheses. It is of course the coproduct (with same notation as the set disjoint union) which is the direct sum for many algebraic categories. – yadaddy Oct 23 '15 at 12:06
• – Martin Sleziak Oct 23 '15 at 12:21
• In particular, notice the comment in the linked question saying that every limit can be obtained as and equalizer of a product. (Assuming the category has equalizers and products, i.e., it is complete.) The construction is described in one part of the proof of Theorem 12.3 in the book by Adámek, Herrlich and Strecker. – Martin Sleziak Oct 23 '15 at 12:27