Here on the page 10, there is a notion of a finitary relation symbol interpretable in a category $\cal K$ with $U:{\cal K}\to \mathbb {Set}$ whose morphisms are monomorphisms preserved by $U$.
Directed colimits preserving subfunctor of $U^n$ is called relation symbol interpretable in $\cal K$.
I do not understand how the canonical functor $G:\cal K\to \mathbb {Str(\Sigma_{\cal K})}$ works.
I also do not follow why $f:GA\to GB$ is iso given $f:UA\to UB$ is iso, and given that $R(f)$ is iso for each subfunctor $R$ of $U^n$. For that $f$ being an iso means that it both reflects and preserves $R$. I do follow why it preserves function symbols.