injectivity

[categories <cat-dist@mta.ca>]

Date: Fri, 9 May 1997 15:30:53 +0200 (MET DST)
From: Marek Golasinski <marek@mat.uni.torun.pl>

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