  Let $R$ be a commutative ring with unit. We develop a Hochschild cohomology
theory in the category $\mathcal{F}$ of linear functors defined from an
essentially small symmetric monoidal category enriched in $R$-Mod, to $R$-Mod.
The category $\mathcal{F}$ is known to be symmetric monoidal too, so one can
consider monoids in $\mathcal{F}$ 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 $\mathcal{F}$ lead to nice
definitions and provide effective tools to prove the main properties and
results of the classical Hochschild cohomology theory.
