I think I get what's going on.
\$$\mathbf{Feas}\$$ is something like a category of pre-ordered sets and *relations* between them. In fact, if we restrict \$$X\$$ and \$$Y\$$ to just sets (with no pre-ordering on them), then \$$\mathbf{Feas} \mid\_{\mathbf{Set}} \cong \mathbf{Rel}\$$

Edit: Here \$$\mathcal{C} \mid\_{\mathcal{D}}\$$ just means that category \$$\mathcal{C}\$$ is restricted to \$$\mathcal{D}\$$.

Thinking about it a bit, I think I could answer **Puzzle 180** maximally efficiently (lazily) by stating that \$$\mathbf{Feas}\$$ is a category if, and only if, the following diagram in \$$\mathbf{Cat}\$$ commutes,

\$\begin{matrix} \mathbf{Set} & \overset{i}\rightarrow & \mathbf{Rel}\\\\ \iota\downarrow & & \downarrow \iota' \\\\ \mathbf{Poset} & \underset{i'}\rightarrow & \mathbf{Feas} \end{matrix} \$

So, if I understand, the composition in \$$\mathbf{Feas}\$$ should be like that of both \$$\mathbf{Rel}\$$, where we compose on an existence of common objects, and that of \$$\mathbf{Poset}\$$, where composition preserves the monotone structure.