  Let $(X,J)$ be an almost-complex manifold. In \cite{li-zhang} Li and Zhang
introduce $H^{(p,q),(q,p)}_J(X)_{\rr}$ as the cohomology subgroups of the
$(p+q)$-th de Rham cohomology group formed by classes represented by real
pure-type forms. Given a proper, surjective, pseudo-holomorphic map between two
almost-complex manifolds we study the relationship among such cohomology
groups. Similar results are proven in the symplectic setting for the cohomology
groups introduced in \cite{tsengyauI} by Tseng and Yau and a new
characterization of the Hard Lefschetz condition in dimension $4$ is provided.
