  We define Schur categories, $\Gamma^d \mathcal C$, associated to a
$\Bbbk$-linear category $\mathcal C$, over a commutative ring $\Bbbk$. The
corresponding representation categories, $\mathbf{rep}\, \Gamma^d\mathcal C$,
generalize categories of strict polynomial functors. Given a
$\Bbbk$-superalgebra $A$, we show that for certain categories $\mathcal{V} =
\boldsymbol{\mathcal V}_A$, $\boldsymbol{\mathcal E}_A$ of $A$-supermodules,
there is a Morita equivalence between $\mathbf{rep}\, \Gamma^d\mathcal{V}$ and
the category of supermodules over a generalized Schur superalgebra of the form
$S^A(m|n,d)$ and $S^A(n,d)$, respectively. We also describe a formulation of
generalized Schur-Weyl duality from the viewpoint of the category
$\mathbf{rep}\, \Gamma^d \boldsymbol{\mathcal E}_A$.
