Representations are adjoint to endomorphisms

The functor that takes a ring to its category of modules has an adjoint if one remembers the forgetful functor to abelian groups: the endomorphism ring of linear natural transformations. This uses the self-enrichment of the category of abelian groups. If one considers enrichments into symmetric sequences or even bisymmetric sequences, one can produce an endomorphism operad or an endomorphism properad. In this note, we show that more generally, given a category enriched in a monoidal category , the functor that associates to a monoid in its category of representations in is adjoint to the functor that computes the endomorphism monoid of any functor with domain . After describing the first results of the theory we give several examples of applications.