Hochschild cohomology for functors on linear symmetric monoidal categories

Let R be a commutative ring with unit. We develop a Hochschild cohomology theory in the category F of linear functors defined from an essentially small symmetric monoidal category enriched in R-Mod, to R-Mod. The category F is known to be symmetric monoidal too, so one can consider monoids in F and modules over these monoids, which allows for the possibility of a Hochschild cohomology theory. The emphasis of the article is in considering natural hom constructions appearing in this context. These homs, , together with the abelian structure of F, lead to nice definitions and provide effective tools to prove the main properties and results of the classical Hochschild cohomology theory.