A Relationship between Left Exact and Representable Functors

Our aim in this paper is to demonstrate a relationship between left exact and representable functors. More precisely, in the functor category whose objects are the additive functors from the dual of an abelian category 𝔄 to the category of abelian groups and whose morphisms are the natural transformations between them, the left exact functors can be characterized as those equivalent to a direct limit of representable functors taken over a directed class. The proof will proceed in the following manner. Lambek [3] and Ulmer [7] have shown that any functor T in can be expressed as a direct limit of representable functors taken over a comma category. When T is left exact, it is easily observed that this comma category is a filtered category. When T is left exact, it is easily observed that this comma category is a filtered category.