Morita equivalences between cyclotomic KLR algebras in types ${\mathtt{C}}_{\mathrm{\infty }}$ and ${\mathtt{A}}_{\mathrm{\infty }}$

We prove that level one cyclotomic KLR algebras in type ${\mathtt{C}}_{\mathrm{\infty }}$ are graded Morita equivalent to level two cyclotomic KLR algebras in type ${\mathtt{A}}_{\mathrm{\infty }}$. We hence deduce the graded decomposition numbers and full submodule structures of all level one cyclotomic KLR algebras in type ${\mathtt{C}}_{\mathrm{\infty }}$.