Let $\left\{\Pi_i\right\}_{i \in I}$ be a family of problems. Let problem $\Pi_i$ have solution $u_i$ lying in some solution space $X_i$. I am interested in making this set into a category. Is it meaningful to define morphisms between problems as follows?
Define a morphism $f_{ij}:\Pi_i \rightarrow \Pi_j$ if there exists a map $\sigma: X_i \rightarrow X_j$ such that $u_j = \sigma(u_i)$.
Any insights would be greatly appreciated.