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 reflective subcategory of $K$, i.e. when does the inclusion functor $\iota:K'\to K$ admit a left adjoint functor?

For example, I was considering monoids as a subcategory of groups. Here the theory of groups is given by $T' = T\cup \{\forall x \exists y\;xy=1\}$, where T is the theory of monoids. As far as I can tell though, the similarly structured sentence $\forall x\exists y\; x\sim y$ does not form a reflective subcategory for graphs, so presumably this cannot be characterized in terms of quantifier complexity, i.e. just being $\Pi_n$ or $\Sigma_n$.