On the object-wise tensor product of functors to modules 

Marek Golasinski 

Theory and Applications of Categories, Vol. 7, 2000, No. 11, pp 226-235

We investigate preserving of projectivity and injectivity by the object-wise 
tensor product of $R\Bbb{C}$-modules, where $\Bbb{C}$ is a small category. 
In particular, let ${\cal O}(G,X)$
be the category of canonical orbits of a discrete group $G$, over a $G$-set 
$X$. We show that projectivity of $R{\cal O}(G,X)$-modules is preserved by 
this tensor product. Moreover, if
$G$ is a finite group, $X$ a finite $G$-set and $R$ is an integral domain 
then such a tensor product of two injective $R{\cal O}(G,X)$-modules is 
again injective.