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.

\\(\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.