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

Is there always a map of operads 
$$\mathbf{End}_{\text{operad}}(Z\times Z)\rightarrow \mathbf{End}_{\text{operad}}(Z) $$ 

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