  The purpose of this article is to study the existence of Deligne's tensor
product of abelian categories by comparing it with the well-known ten- sor
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 Deligne tensor product does not exist is given.
