# left adjoint preserve join applicable on empty set?

In book: https://math.mit.edu/~dspivak/teaching/sp18/7Sketches.pdf 2.87(b) & answer of 2.104(1) assume that left adjoint preserve join applicable on empty set, i.e. $$\left( v \otimes \bigvee_\left(a \in A \right)a\right) \cong \bigvee_\left(a \in A\right)(v \otimes a)$$ where $$A \subseteq V$$.

I think it's not applicable on $$\emptyset$$, as counter example: $$\left(\left[1, \infty\right], \le, \times, \div\right)$$ is symmetric monoidal preorder that is closed.
But $$2 \times \bigvee_\left(a \in \emptyset\right) a = 2 \neq 1 = \bigvee_\left( a \in \emptyset\right) (2 \times a )$$
• I don't see that this monoidal structure is closed, because your "Hom" operation, division, can produce numbers smaller than $1$. – Andreas Blass Apr 14 at 3:27