ct.category theory - In what cases does a Yoneda-like embedding preserve monoidal structure?

Day showed that, for suitable V, any monoidal structure on a (V-)functor category $[C^{mathrm{op}}, V]$ is essentially determined by its restriction to the representables as
$$ F otimes G = int^{A,B} F A otimes G B otimes P(A,B,-) $$
where $P(A,B,-) = C(-, A) otimes C(-, B)$ is a profunctor $C otimes C otimes C^{mathrm{op}} to V$. P (together with a unit and the usual structural isos) is said to endow C with a promonoidal structure.

If C is already a monoidal V-category, then there is a canonical promonoidal structure on it given by
$$ C(-, A) otimes C(-, B) = C(-, A otimes B) $$
In that case, the Yoneda embedding is strong monoidal by definition. In fact it is the unit for the monoidal cocompletion of C.