left adjoint preserve join applicable on empty set?

Question

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 $A \subseteq V$.

Answer 1

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 )$

Please advise if I have any misunderstanding here?

Answer 2

Yes, the initial object is the empty colimit and is thus preserved by all left adjoints. In the context of a poset this says that all left adjoints preserve the bottom element.