Consider an elementary class, $K$, of some $\mathcal{L}$-theory, $T$ equipped with the usual $\mathcal{L}$-structure homomorphisms. (Not elementary embeddings, which elementary classes are more frequently equipped with.) Suppose we have $K' \subseteq K$, the elementary class of models of $T'$. When is $K'$ a reflexive subcategory of $K$?

For example, I was considering monoids as a subcategory of groups. Here $T' = T\cup \{\forall x \exists y\;xy=1\}$, but as far as I can tell the similarly structured sentence $\forall x\exists y\; x\sim y$ does not form a reflexive subcategory for graphs, so presumably this cannot be characterized in terms of just being $\Pi_n$ or $\Sigma_n$.