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 
$$\mathbf{End}_{operad}(X\times Y)\rightarrow \mathbf{End}_{operad}(X) $$ 

Edit: Intuitively, I would say the answer should be NO i.e. there is NOT ALWAYS such a map of operad. But I don't have a concrete counterexample.