Dear Colleagues, Let $Vect_k$ be the category of vector spaces over a field $k$ and $I$ a small category. Consider the category $I-Vect_k$ of all covariant functors from $I$ to $Vect_k$. For two object $F,F'$ of the category $I-Vect_k$ consider their tensor product $F\otimes F'$ such that $(F\otimes F')(i)=F(i)\otimes F'(i)$ for all $i\in I$ and in the obvious way on the morphisms of $I$. 1) Is it true that this tensor product $F\otimes F'$ is injective provided that $F$ and $F'$ are injective? I am really intersted in its particular case. Namely, let $G$ be a finite group and $O(G)$ the finite associated category of canonical orbits. Objects of $O(G)$ are given by the finite $G$-sets $G/H$ for all subgroups $H\subsetq G$ and morphisms by eqivariant maps. 2) What about preserving the injectivity by the above defined tensor product in the functor category $O(G)-Vect_k$? If that is not true for $I=O(G)$ then I would greatly appreciate getting a counterexample. Many thanks in advance for your kind attention on the problem above. With my best regards, Marek Golasinski