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!