Exact Functor - Generalization

Generalization

In SGA4, tome I, section 1, the notion of left (right) exact functors are defined for general categories, and not just abelian ones. The definition is as follows:

Let C be a category with finite projective (resp. inductive) limits. Then a functor u from C to another category C′ is left (resp. right) exact if it commutes with projective (resp. inductive) limits.

Despite its abstraction, this general definition has useful consequences. For example, in section 1.8, Grothendieck proves that a functor is pro-representable if and only if it is left exact, under some mild conditions on the category C.