Models of generalized algebraic theories

For any finite limit category $\mathsf{E}$ and GAT $T$, there is a category of models of $T$ in $\mathsf{E}$

Under certain conditions (for instance if $\mathsf{E}$ is a topos), the category of models has an initial object, which is called the term model. We can use this to build models.

Example. Consider the initial model $P$ of

$(G:\mathsf{ReflGraph})\times (x:G.V)\times (y:G.V)\times (f:G.Exy)\times (g:G.Eyx)$

Then $P.G$ is the graph