Comparison of canonical bases for Schur and universal enveloping algebras

We show that canonical bases in $\dot{U}(\mathfrak{sl}_n)$ and the Schur algebra are compatible; in fact we extend this result to $p$-canonical bases. This follows immediately from a fullness result from a functor categorifying this map. In order to prove this result, we also explain the connections between categorifications of the Schur algebra which arise from parity sheaves on partial flag varieties, singular Soergel bimodules and Khovanov and Lauda's "flag category," which are of some independent interest.