(Notice that the latter implies the former, since for every object there is the identity morphism on that object). Finite categories form the 2-category, FinCat.

Similarly, a locally finite category is a category enriched over $Fin Set$, that is a category whose hom-sets are all finite.

One is often interested in whether an arbitrary category $D$ has limits and colimits indexed by finite categories. A category with all finite limits is called finitely complete or left exact (or lex for short). A category with all finite colimits is called finitely cocomplete or right exact.