Thanks Marius, Tobias, and John for the responses! I had a lot of fun working on these problems.

Marius, I really like the example of getting a discount instead of a bag charge! Your comment about opposite categories also made me think that given a function \$$f: X \to Y\$$ we can define a relation on \$$X\$$ in an opposite way by \$x \leq_X x' \iff f(x) \geq_Y f(x').\$

I also wanted to check my thinking about Puzzle 77 again.

I showed that the property \$$f(x) \otimes_Y f(x') = f(x \otimes_X x') \$$ is sufficient for making \$$(X, \leq_X, \otimes_X, 1_x) \$$ a monoidal pre-order. But the examples of a bag cost and coupon discount that Marius and I suggested, show that this is not a necessary condition, since in both of those cases we only have \$$f(x) \otimes_Y f(x') \leq f(x \otimes_X x') \$$ and \$$f(x) \otimes_Y f(x') \geq f(x \otimes_X x') \$$ respectively. So as of yet, we don't have a nice necessary and sufficient condition on \$$f\$$ for making \$$(X, \leq_X, \otimes_X, 1_x) \$$ a monoidal pre-order. Is that correct?