What kinds of Yoneda-like situations induce an embedding that preserves the tensor product for some arbitrary monoidal category?


The cases where the monoidal product is given by a limit or colimit give this immediately for the usual Yoneda embedding, but this breaks down for "real" monoidal categories like $(Vect, \otimes)$.


Are there $V$-enriched cases where the generalised embedding


$$ Y : C \to V^{C^{op}} $$


does preserve the tensor product for interesting monoidal categories $C$?



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.


