I'm sad that nobody has done Puzzle 180:

> **Puzzle 180.** Prove that there is a category \\(\textbf{Feas}\\) whose objects are preorders and whose morphisms are feasibility relations, composed as above.

It's really important and it's also not hard. Perhaps the word "prove" is scary to some people? This proof is mostly just a calculation. The main thing to check is that composition is _associative_. So, assume we've got feasibility relations

\[ F: W \nrightarrow X, \; G : X \nrightarrow Y, \; H: Y \nrightarrow Z .\]

We want to compute

\[ (HG) F \]

and

\[ H(G F) \]

and show they are equal. We know the formula for composition. So, _dive in!_

Checking that our would-be category \\(\mathbf{Feas}\\) has identity morphisms requires a bit more creativity, because we need to _guess_ what the identity morphisms _are_ before checking that they work.

But as usual in category theory, this guessing is easy because there's so little to work with - any answer you can write down has a chance of making sense is likely to be right. We have a preorder \\(X\\) and we want to cook up a feasibility relation from \\(X\\) to itself. _What can we possibly do???_

> **Puzzle 180.** Prove that there is a category \\(\textbf{Feas}\\) whose objects are preorders and whose morphisms are feasibility relations, composed as above.

It's really important and it's also not hard. Perhaps the word "prove" is scary to some people? This proof is mostly just a calculation. The main thing to check is that composition is _associative_. So, assume we've got feasibility relations

\[ F: W \nrightarrow X, \; G : X \nrightarrow Y, \; H: Y \nrightarrow Z .\]

We want to compute

\[ (HG) F \]

and

\[ H(G F) \]

and show they are equal. We know the formula for composition. So, _dive in!_

Checking that our would-be category \\(\mathbf{Feas}\\) has identity morphisms requires a bit more creativity, because we need to _guess_ what the identity morphisms _are_ before checking that they work.

But as usual in category theory, this guessing is easy because there's so little to work with - any answer you can write down has a chance of making sense is likely to be right. We have a preorder \\(X\\) and we want to cook up a feasibility relation from \\(X\\) to itself. _What can we possibly do???_