Endomorphism category of an abelian category

Let 𝒞 be an additive category. Denote by End(𝒞) the endomorphism category of 𝒞, i.e., the objects in End(𝒞) are pairs (C,c) with C∈𝒞,c∈End_𝒞(C), and a morphism f:(C,c)→(D,d) is a morphism f∈Hom_𝒞(C,D) satisfying fc?=?df. This paper is devoted to an approach of the general theory of the endomorphism category of an arbitrary additive category. It is proved that the endomorphism category of an abelian category is again abelian with an induced structure without nontrivial projective or injective objects. Furthermore, the endomorphism category of any nontrivial abelian category is nonsemisimple and of infinite representation type. As an application, we show that two unital rings are Morita equivalent if and only if the endomorphism categories of their module categories are equivalent.