Simple semigroups in finite categories

Abstract: In this paper, we classify finite categories with two objects such that one of the endomorphism monoids is a group. We prove that having a group on one side affects the structure of the other endomorphism monoid, and we prove that it is going to contain a simple semigroup. We also prove the other direction, that if we have a Rees matrix semigroup we can construct a category with two objects such that one of the objects is a group.