  Using the affine web category introduced in a prequel as a building block, we
formulate a diagrammatic $\Bbbk$-linear monoidal category, the affine Schur
category, for any commutative ring $\Bbbk$. We then formulate diagrammatic
categories, the cyclotomic Schur categories, with arbitrary parameters at
positive integral levels. Integral bases consisting of elementary diagrams are
obtained for affine and cyclotomic Schur categories. A second diagrammatic
basis, called a double SST basis, for any such cyclotomic Schur category is
also established, leading to a conjectural higher level RSK correspondence. We
show that the endomorphism algebras with the double SST bases are isomorphic to
degenerate cyclotomic Schur algebras with their cellular bases, providing a
first diagrammatic presentation of the latter. The presentations for the affine
and cyclotomic Schur categories are much simplified when $\Bbbk$ is a field of
characteristic zero.
