# Normalization of profunctors over a monoidal category

Let $(ℂ,⊗,I)$ be a monoidal category. The category of endoprofunctors $ℂ^{op} × ℂ → \mathbf{Set}$ is duoidal with composition $(◁)$ and Day convolution $(⊛)$.
$$(ℂ^{op}×ℂ,\mathbf{Set}, ⊛, I, ◁, \hom).$$
The category of $\hom^⊛$-bimodules has been traditionally called the category of Tambara modules. Their tensoring is possible because $[ℂ{op}×ℂ,\mathbf{Set}]$ admits reflexive coequalisers.
$$\mathcal{N}(ℂ^{op}×ℂ,\mathbf{Set}, ⊛, I, ◁, \hom) = (\mathbf{Tamb}, ⊛_{\hom}, \hom, ◁, \hom).$$**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).