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).