Tensor products of finitely cocomplete and abelian categories

The purpose of this article is to study the existence of Deligneʼs tensor product of abelian categories by comparing it with the well-known tensor product of finitely cocomplete categories. The main result states that the former exists precisely when the latter is an abelian category, and moreover in this case both tensor products coincide. An example of two abelian categories whose Deligneʼs tensor product does not exist is given.