Category of models of an algebraic theory using "models" of its Cauchy completion

Let $\mathcal{T}$ be an algebraic theory (small category with finite products) and $\bar{\mathcal{T}}$ be its Cauchy completion.

What kind of functors (objects) yield a full subcategory of $\mathsf{Set}^{\bar{\mathcal{T}}}$ equivalent to the category of models $\operatorname{Mod} \mathcal T$ of $\mathcal T$ in $\mathsf{Set}$?