Algebra objects of $\operatorname{Vec}(\mathscr{C})$ are lax functors $\mathscr{C}^\text{op}\to \operatorname{Vec}$

Question

Let $\mathscr{C}$ be a rigid $C^*$-tensor category. Let $\operatorname{Vec}(\mathscr{C})$ be the category with linear functors $\mathscr{C}^{\text{op}}\to \operatorname{Vec}$ (= category of complex vector spaces) as objects and with natural transformations between these functors as morphisms. This is a monoidal category, see section 2.4 in the article "Operator algebras in rigid C*-tensor categories" by Jones and Penneys.

In this same article, the following is claimed:

I'm now trying to verify this claim. In neither direction, I see how this correspondence should work. So concretely, given an algebra object in $\operatorname{Vec}(\mathscr{C})$, how to associate a lax tensor functor $\mathscr{C}^{\text{op}}\to \operatorname{Vec}$ and conversely?

Thanks in advance for your help!

Answer 1

This is a special case of the well known result in (enriched) category theory which I believe goes back to Day, for example see Proposition 3.4 of https://ncatlab.org/nlab/show/Day+convolution#Monoids. The enriching category V utilized in the paper you reference is the symmetric monoidal category of vector spaces.

Answer 2

A somewhat more explicit answer. Details are still to be filled in, but at least you get an idea how to go from one to the other:

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If $\mathbf{A}$ is an algebra object in $\operatorname{Vec}(\mathcal{C})$, then we have for every object $c$ maps $$\bigoplus_{a,b} \mathbf{A}(a)\otimes \mathcal{C}(c, a\otimes b)\otimes \mathbf{A}(b) \to \mathbf{A}(c)$$ where the direct sum ranges over simple objects. Taking $x,y$ simple, we consider the composition $$\mathbf{A}(x)\otimes \mathbf{A}(y)\hookrightarrow \bigoplus_{a,b} \mathbf{A}(a)\otimes \mathcal{C}(x\otimes y, a\otimes b)\otimes \mathbf{A}(b) \to \mathbf{A}(x\otimes y).$$ This gives us morphisms $\mathbf{A}(x)\otimes \mathbf{A}(y)\to \mathbf{A}(x\otimes y)$. There is a unique natural transformation between the bilinear functors $\otimes \circ (\mathbf{A}\times \mathbf{A})\to \mathbf{A}\circ \otimes$ with these values on the simple objects. This gives the desired Lax-tensor structure.

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Conversely, if $\mathbf{A}: \mathcal{C}^{\text{op}}\to \operatorname{Vec}$ is a lax functor, with maps $\mathbf{A}(x)\otimes \mathbf{A}(y)\to \mathbf{A}(x\otimes y)$, consider the composition $$\bigoplus_{a,b}\mathbf{A}(a)\otimes \mathcal{C}(c, a\otimes b)\otimes \mathbf{A}(b) \cong \bigoplus_{a,b}\mathbf{A}(a\otimes b) \otimes \mathcal{C}(c,a\otimes b) \to \mathbf{A}(c)$$ which gives maps $(\mathbf{A}\otimes \mathbf{A})(c)\to \mathbf{A}(c)$ which induce the multiplication $\mathbf{A}\otimes \mathbf{A} \implies\mathbf{A}$.