The existence of abelian categories with a finite number of objects or with finite Homs is investigated. Thus a complete characterization of categories having a finite number of objects and morphisms, with products and a zero-object, is given. The non existence of abelian categories with a finite number of objects, non finite Hom and with endomorphisms satisfying the following condition is then proved: every endomorphismis a mono-epimorphism. An example of abelian category with a unique non zero-object and non finite Hom is finally obtained.