Normalization of profunctors over a monoidal category

Let $(C,\otimes ,I)$ be a monoidal category. The category of endoprofunctors $C^{op}\times C\to \mathbf{Set}$ is duoidal with composition $(◁)$ and Day convolution $(⊛)$. $(C^{op}\times C,\mathbf{Set},⊛,I,◁,\mathrm{hom}).$ The category of ${\mathrm{hom}}^{⊛}$-bimodules has been traditionally called the category of Tambara modules. Their tensoring is possible because $[Cop\times C,\mathbf{Set}]$ admits reflexive coequalisers. $\mathcal{N}(C^{op}\times C,\mathbf{Set},⊛,I,◁,\mathrm{hom})=(\mathbf{Tamb},{⊛}_{\mathrm{hom}},\mathrm{hom},◁,\mathrm{hom}).$Theorem. The category of Tambara modules is a normal duoidal category and, in fact, it is the normalization of the duoidal category of endoprofunctors.

Normalization is the process that transforms wires into boards, see Tambara modules as algebras.

Tags: Tambara module, duoidal normalization.

References: I believe this result, in this formulation, is novel. For reference on the normalization of a duoidal category, see Commutativity (Garner, Lopez Franco, 2015).