This question relates to my previous question found here: Defining Category of Problems
Let $\left\{\Pi_i \right\}_{i \in I}$ be a family of problems. Let the solution $u_i$ of $\Pi_i$ lie in some space $X_i$. I want to make this family into a category in order to study decomposition of problems into products of smaller ones. I want to define the morphisms of this category in a manner that, the existence of a morphism $f_{j}^i : \Pi_i \rightarrow \Pi_j$ is equivalent to saying that solving $\Pi_i$ implies solving $\Pi_j$.
(from here and onwards re-edited)
I define a morphism $f_{j}^i : \Pi_i \rightarrow \Pi_j$ to be a map $f: X_i \rightarrow X_j$ such that $f(u_i) = u_j$ and $f$ does not depend on $u_j$.
Question: does this definition capture the interpretation of morphisms that i want? I.e. let $f_{j}^i : \Pi_i \rightarrow \Pi_j$ be a morphism. Does it follow that solving $\Pi_i$ implies solving $\Pi_j$?
thanks :-)