An $R$ -category $A$ is a category equipped with an $R$ -module structure on each hom set such that the composition is $R$ -bilinear. More precisely, let us assume for instance that we are given a commutative ring $R$ with identity. Then a small $R$ -category-or equivalently an $R$ -algebroid- will be defined as a category enriched in the monoidal category of $R$ -modules, with respect to the monoidal structure of tensor product. This means simply that for all objects $b,c$ of $A$ , the set $A(b,c)$ is given the structure of an $R$ -module, and composition $A(b,c) \times A(c,d) \lra A(b,d)$ is $R$ -bilinear, or is a morphism of $R$ -modules $A(b,c) \otimes_R A(c,d) \lra A(b,d)$ .