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We use duality theorems to obtain presentations of some categories ofmodules. To derive these presentations we generalize a result ofCautis-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 $V$, ifevery 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 conditionsfor the existence of an isomorphism of categories between the full subcategoryof $\mathcal{R}ep A$ whose objects are $\mathfrak{g}$-weight spaces of $P$, anda quotient of the category version of Lusztig's idempotented form,$\dot{{\mathcal{U}}} \mathfrak{g}$, formed by setting to zero all morphismsfactoring through a collection of objects in $\dot{{\mathcal{U}}} \mathfrak{g}$depending on $P$. This is essentially a categorical version of theidentification of generalized Schur algebras with quotients of Lusztig'sidempotented forms given by Doty in [arXiv:math/0305208].  Applied to Schur-Weyl Duality we obtain a diagrammatic presentation of thefull subcategory of $\mathcal{R}ep S_d$ whose objects are direct sums ofpermutation modules, as well as an explicit description of the$\otimes$-product of morphisms between permutation modules. Applied toBrauer-Schur-Weyl Duality we obtain diagrammatic presentations of subcategoriesof $\mathcal{R}ep \mathcal{B}_{d}^{(- 2n)}$ 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.

Presentations of Categories of Modules using the Cautis-Kamnitzer-Morrison Principle