  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.
