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 $\mathsf{Set}$?



Category of models of an algebraic theory using "models" of its Cauchy completion
Because $\bar{\mathcal{T}}$ is an algebraic theory *itself* we have:


$$\operatorname{Mod} \mathcal T \simeq \operatorname{Mod} \bar{\mathcal T}$$


because $\mathcal{T}$ and $\bar{\mathcal T}$ have the same Cauchy completion.


Taken from Adameck et al "Algebraic Theories" Chapter 15.


