# Examples/Papers where a monoidal structure restricts to a promonoidal structure on a dense full subcategory

In Brian Day's thesis, he gives a definition of a pro-monoidal structure based on a particular motivating example:

Suppose $M:\cal{A}\hookrightarrow \cal{B}$ is the inclusion of a small full subcategory, and suppose $\cal{B}$ is equipped with a monoidal structure $1\in \cal{B}$ and $\otimes:\cal{B} \times \cal{B} \to \cal{B}$. Summarizing the discussion, he essentially gives a condition for the monoidal structure on $\cal{B}$ to restrict to a pro-monoidal structure on $\cal{A}$, which amounts to showing that the four induced maps on coends:

$$\int^X B(MX,1) \times B(M(-), MX\otimes MA) \\ \downarrow \\ \int^Y B(Y,1) \times B(M(-), Y\otimes MA)$$

$$\int^X B(MX,1) \times B(M(-), MA\otimes M(X)) \\ \downarrow \\ \int^Y B(Y,1) \times B(M(-), MA\otimes Y)$$

$$\int^X B(MX,MA\otimes MA') \times B(M(-), MX\otimes MA'') \\ \downarrow \\ \int^Y B(Y,MA\otimes MA') \times B(M(-), Y\otimes MA'')$$

$$\int^X B(MX,MA'\otimes MA'') \times B(M(-), MA\otimes MX) \\ \downarrow \\ \int^Y B(Y,MA'\otimes MA'') \times B(M(-), MA\otimes Y)$$

are isomorphisms for all $A,A',A''\in \cal{A}$. (I've used the opposite variance for the Yoneda reduction to the one in Day's paper). Morally this seems to say that when we restrict to tensor products of objects of $\cal{A}$, it is enough to compute the coend over $\cal{A}$ rather than all of $\cal{B}$.

In the situation that I care about, $\cal{A}$ is actually also dense in $\cal{B}$, the monoidal structure on $\cal{B}$ is biclosed, and the unit of the monoidal structure lies in $\cal{A}$ and is the terminal object of both $\cal{A}$ and $\cal{B}$, so the first two maps seem to end up working out to be isomorphisms pretty easily (I haven't proven it, but I've convinced myself that it works).

I have two questions then:

1.) If $\cal{A}$ is also dense in $\cal{B}$, can these conditions be simplified in any way? What about if any of the other conditions relevant to my case of interest hold ($\cal{B}$ biclosed or $1\in \cal{A}$ and terminal in both categories)?

2.) Are there any examples in the literature where the existence of the induced pro-monoidal structure on $\cal{A}$ has been proven by using these coend maps or a related technique (specifically in the nontrivial case where $\cal{B}$ is not just the category of presheaves on $\cal{A}$ and $\cal{A}$'s induced pro-monoidal structure is not representable).