This looks great, Matthew! Puzzle 199 would have been hard for many students because the "workhorses" required to solve this puzzle are the distributive laws:

$a \otimes \left( \bigvee\_{b\in B} b\right) = \bigvee\_{b \in B} (a \otimes b) \quad ; \quad \bigvee\_{b\in B} (b \otimes a) = \left(\bigvee\_{b \in B} b\right) \otimes a$

which hold for every element \$$a\$$ and every subset \$$B\$$ of a commutative quantale \$$\mathcal{V}\$$. The problem is that these need to be proved starting with the definition:

**Definition.** A **quantale** \$$\mathcal{V}\$$ is a closed monoidal poset that has all joins.

The solution, as you so deftly note, is that \$$\mathcal{V} \\$$ being closed implies that the operation \$$a \otimes - \$$ is a left adjoint, so it preserves all joins. And first distributive law above says precisely that \$$a \otimes - \$$ preserves all joins!

The second one says that \$$- \otimes a\$$ preserves all joins, but in a _commutative_ quantale we have \$$- \otimes a = a \otimes - \$$.

Those who are puzzled can look back at [Lecture 16](https://forum.azimuthproject.org/discussion/2031/lecture-16-chapter-1-the-adjoint-functor-theorem-for-posets/p1), where we proved this:

**Theorem.** If a monotone function \$$f: X \to Y \$$ is a left adjoint, it preserves joins whenever they exist. That is, whenever a set \$$S \subseteq X \$$ has a join we have

$f (\bigvee S ) = \bigvee \\{ f(x) : \; x \in S\\} .$

But the conditions of 'preserving joins' and 'being a left adjoint' are more closely connected that this theorem acknowledges. In [Lecture 16](https://forum.azimuthproject.org/discussion/2031/lecture-16-chapter-1-the-adjoint-functor-theorem-for-posets/p1) we went on to prove this:

**Adjoint Functor Theorem for Posets.** Suppose \$$X\$$ is a poset that has all joins and \$$Y \$$ is any poset. Then a monotone map \$$f : X \to Y \$$ is a left adjoint if and only if it preserves all joins.

Using this, we see that if \$$\mathcal{V}\$$ is a monoidal poset with all joins, and the operation \$$a \otimes -\$$ preserves joins, it must be a left adjoint! So, we get this nice result, which is a special case of Proposition 2.70 in _[Seven Sketches](http://math.mit.edu/~dspivak/teaching/sp18/7Sketches.pdf)_:

**Theorem.** If \$$\mathcal{V}\$$ is a monoidal poset with all joins, \$$\mathcal{V}\$$ is a quantale if and only if

$a \otimes \left( \bigvee\_{b\in B} b\right) = \bigvee\_{b \in B} (a \otimes b)$

for every element \$$a\$$ and every subset \$$B\$$ of \$$\mathcal{V}\$$.

So, to help out future students, I'll add some of this material to the lecture!