If a module $M$ over a ring $R$ has a finite presentation (i.e. is the quotient of a module of finite type by a submodule which is also of finite type), then the functor $Hom_R(M,-)$ commutes with filtered colimits. The converse is true: if the functor $Hom_R(M,-)$ preserves filtered colimits, then $M$ has a finite presentation.

Remark: The possibilty of determining an object through finitely many generators and relations can very often characterized by the property the taking maps out of it is compatible with filtered colimits/unions (e.g. for groups, rings, and so on). Taking the property that $Hom(M,-)$ preserves filtered colimits as a definition is the notion of *object of finite presentation* in any category, which make sense even if there are no generators nor relations to discuss. Studying categories which are generated by objects of finite presentation is thus a natural thing to do. Replacing filtered diagrams by $\kappa$-filtered ones for various cardinals $\kappa$, this naturally leads to the theory of presentable categories, which is a major branch of category theory (being in the background of the theory of Grothendieck topoi, for instance), so robust that is has a counterpart in $\infty$-category theory. Developping this kind of ideas in a derived/homotopical context has also proved to be very useful (e.g. the notion of prefect complex of quasi-coherent sheaves).