Let me recall that any small category $\mathbb{A}$ enriched in a complete and cocomplete symmetric monoidal closed category $\mathbb{V}$ admits embedding (the Yoneda embedding):
$$y\_\mathbb{A} \colon \mathbb{A} \rightarrow \mathbb{V}^{\mathbb{A}^{op}}$$
into a complete and cocomplete $\mathbb{V}$-enriched category. And:

* this embedding preserves any monoidal structure $\langle \otimes\_\mathbb{A}, I\_\mathbb{A} \rangle$ defined on $\mathbb{A}$; the induced structure $\mathbb{V}^{\mathbb{A}^{op}} \otimes \mathbb{V}^{\mathbb{A}^{op}} \rightarrow \mathbb{V}^{\mathbb{A}^{op}}$ is given by the Day convolution:
$$\langle F, G \rangle \mapsto \int^{B \in \mathbb{A}, C \in \mathbb{A}} F(B) \otimes G(C) \otimes \hom(-, B \otimes\_\mathbb{A} C)$$
* the induced structure on $\mathbb{V}^{\mathbb{A}^{op}}$ is always monoidal (bi)closed
* more generally, any *promonoidal* structure (i.e. a weak monoidal object in the bi-category of $\mathbb{V}$-enriched distributors) defined on $\mathbb{A}$ induces a (bi)closed monoidal structure on $\mathbb{V}^{\mathbb{A}^{op}}$; moreover there is a bijective correspondence between (bi)closed monoidal structures defined on $\mathbb{V}^{\mathbb{A}^{op}}$ and promonoidal structures defined on $\mathbb{A}$

It seems (though I have not checked details) that all of the above carry to the context of categories *internal* to any finitely cocomplete locally cartesian closed category $\mathbb{C}$. For any $\mathbb{C}$-internal category $\mathbb{A}$ there exists a fibred fully faithful embedding:
$$y\_\mathbb{A} \colon \mathbb{A} \rightarrow \mathbb{C}^{\rightarrow^{\mathbb{A}^{op}}}$$
where $\mathbb{C}^{\rightarrow^{\mathbb{A}^{op}}}$ is the (complete and cocomplete cartesian closed) fibration of $\mathbb{A}^{op}$-indexed diagrams in the fundamental (i.e. codomain) fibration on $\mathbb{C}$.

This observation is so blatantly obvious that it must have been made (and written somewhere) before. What are the references?

Much of the work has been done in "Cosmoi of Internal Categories" by Ross Street. However, I have not found any statement about transporting monoidal structures from $\mathbb{A}$ to $\mathbb{C}^{\rightarrow^{\mathbb{A}^{op}}}$ in the paper.