Homomorphisms between pure mapping class groups

Let $S$ and $S^{\mathrm{\prime }}$ be orientable finite-type surfaces of genus $g\ge 4$ and $g^{\mathrm{\prime }}$, respectively. We prove that every multitwist-preserving map between pure mapping class groups $\text{PMap}(S)\to \text{PMap}(S^{\mathrm{\prime }})$ is induced by a multi-embedding. As an application, we classify all homomorphisms $\text{PMap}(S)\to \text{PMap}(S^{\mathrm{\prime }})$ for $g\ge 4$ and $g^{\mathrm{\prime }}\le 6\cdot 2^{g-4}$.