Let $\mathbf{Top}$ the category of (nice) topological spaces. For any space $Z$, define $\mathbf{End}_{operad}(Z)$ as the endomorphism operad.

Given two spaces $X,Y$, is there always a map of operad $$\mathbf{End}_{operad}(X\times Y)\rightarrow \mathbf{End}_{operad}(X) $$