It is well known that the notion of an internal category can be generalized to categories without pullbacks by considering cotensors of comodules in a monoidal category. I'm curious about the other direction, considering internal monoidal categories in a category with products.
Definition. Let $\mathcal{C}$ be a category with finite products. A strict internal monoidal category $\mathscr{C}$ in $\mathcal{C}$ consists of an internal category $\mathscr{C}$ in $\mathcal{C}$ together with internal functors $\otimes:\mathcal{C}\times\mathcal{C}\to\mathcal{C}$, $I:{\bf 1}\to\mathscr{C}$ such that $$ \otimes\circ(\otimes\times1)=\otimes\circ(1\times\otimes), $$ $$ \otimes\circ\langle1,I\circ!\rangle=1_\mathscr{C}=\otimes\circ\langle I\circ!,1\rangle. $$
We can internalize weak monoidal categories in a bicategory:
Definition. Let $\mathcal{C}$ be a bicategory with products. An internal monoidal category $\mathscr{C}$ in $\mathcal{C}$ consists of an internal category $\mathscr{C}$ in $\mathcal{C}$ together with internal functors $\otimes:\mathcal{C}\times\mathcal{C}\to\mathcal{C}$, $I:{\bf 1}\to\mathscr{C}$ together with internal natural isomorphisms $$ \alpha:\otimes\circ(\otimes\times1)\Rightarrow\otimes\circ(1\times\otimes), $$ $$ \iota^r:\otimes\circ\langle1,I\circ!\rangle\Rightarrow 1_\mathscr{C}, $$ $$ \iota^\ell:\otimes\circ\langle I\circ!,1\rangle\Rightarrow 1_\mathscr{C}, $$ such that the following diagrams commute:
(top is the pentagon identity, bottom is the triangle identity)
Have these notions been studied in the literature anywhere? We can consider an internal braiding in a bicategory by considering an internal natural isomorphism $B^\otimes:\otimes\Rightarrow\otimes\circ\pi_1\times\pi_0$ such that the following diagrams commute
The braiding is symmetric iff ${B^\otimes}^{-1}=B^\otimes_{\pi_1\times\pi_0}$, so on and so forth. I can't find mention of these notions after a few months of playing around with them but I'm certain they've been written down before, I'm just wondering if there's anything that's been made publicly available. Any assistance is appreciated.
