  We use duality theorems to obtain presentations of some categories of
modules. To derive these presentations we generalize a result of
Cautis-Kamnitzer-Morrison [arXiv:1210.6437v4]:
  Let $\mathfrak{g}$ be a reductive Lie algebra, and $A$ an algebra, both over
$\mathbb{C}$. Consider a $(\mathfrak{g} , A)$-bimodule $P$ in which
  (a) $P$ has a multiplicity free decomposition into irreducible $(\mathfrak{g}
, A)$-bimodules.
  (b) $P$ is "saturated" i.e. for any irreducible $\mathfrak{g}$-module $V$, if
every weight of $V$ is a weight of $P$, then $V$ is a submodule of $P$.
  We show that statements (a) and (b) are necessary and sufficient conditions
for the existence of an isomorphism of categories between the full subcategory
of $\mathcal{R}ep A$ whose objects are $\mathfrak{g}$-weight spaces of $P$, and
a quotient of the category version of Lusztig's idempotented form,
$\dot{{\mathcal{U}}} \mathfrak{g}$, formed by setting to zero all morphisms
factoring through a collection of objects in $\dot{{\mathcal{U}}} \mathfrak{g}$
depending on $P$. This is essentially a categorical version of the
identification of generalized Schur algebras with quotients of Lusztig's
idempotented forms given by Doty in [arXiv:math/0305208].
  Applied to Schur-Weyl Duality we obtain a diagrammatic presentation of the
full subcategory of $\mathcal{R}ep S_d$ whose objects are direct sums of
permutation modules, as well as an explicit description of the
$\otimes$-product of morphisms between permutation modules. Applied to
Brauer-Schur-Weyl Duality we obtain diagrammatic presentations of subcategories
of $\mathcal{R}ep \mathcal{B}_{d}^{(- 2n)}$ and $\mathcal{R}ep
\mathcal{B}_{r,s}^{(n)}$ whose Karoubi completion is the whole of
$\mathcal{R}ep \mathcal{B}_{d}^{(- 2n)}$ and $\mathcal{R}ep
\mathcal{B}_{r,s}^{(n)}$ respectively.
