# Finite presentation for left-exact endofunctors

For a commutative unital ring R and a left-exact endofunctor L on the category of R-modules, what is, or should be, the definition of finite presentation for L ? Possibilities:

L( R ) is finitely presented over R .

For every R-algebra S , L( S ) is finitely presented over S .

For some category C (but which one?) having L as an object, closed under filtered colimits, the structural mapping colim Hom( L , M_a ) -> Hom( L , colim M_a ) is bijective for every filtered diagram { M_a } of C-objects.