  In this paper, we show an isomorphism of homological knot invariants
categorifying the Reshetikhin-Turaev invariants for $\mathfrak{sl}_n$. Over the
past decade, such invariants have been constructed in a variety of different
ways, using matrix factorizations, category $\mathcal{O}$, affine
Grassmannians, and diagrammatic categorifications of tensor products.
  While the definitions of these theories are quite different, there is a key
commonality between them which makes it possible to prove that they are all
isomorphic: they arise from a skew Howe dual action of $\mathfrak{gl}_\ell$ for
some $\ell$. In this paper, we show that the construction of knot homology
based on categorifying tensor products (from earlier work of the second author)
fits into this framework, and thus agrees with other such homologies, such as
Khovanov-Rozansky homology. We accomplish this by categorifying the action of
$\mathfrak{gl}_\ell\times \mathfrak{gl}_n$ on
$\bigwedge\nolimits^{\!p}(\mathbb{C}^\ell\otimes \mathbb{C}^n)$ using
diagrammatic bimodules. In this action, the functors corresponding to
$\mathfrak{gl}_\ell$ and $\mathfrak{gl}_n$ are quite different in nature, but
they will switch roles under Koszul duality.
