  We introduce categories of homogeneous strict polynomial functors,
$\Pol^\I_{d,\k}$ and $\Pol^\II_{d,\k}$, defined on vector superspaces over a
field $\k$ of characteristic not equal 2. These categories are related to
polynomial representations of the supergroups $GL(m|n)$ and Q(n), respectively.
In particular, we prove an equivalence between $\Pol^\I_{d,\k}$,
$\Pol^\II_{d,\k}$ and the category of finite dimensional supermodules over the
Schur superalgebra $\Sc(m|n,d)$, $\Qc(n,d)$ respectively provided $m,n \ge d$.
We also discuss some aspects of Sergeev duality from the viewpoint of the
category $\Pol^\II_{d,\k}$.
