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](https://ncatlab.org/nlab/show/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 $\Sigma\_n$.



When is an elementary subclass reflective?
This is a great question, which I will only partially address. A complete, general, answer to the question goes beyond the energy I am happy to put into MathOverflow.


**Def**. Let $T \subset T'$ be first order theories (having models with at least two elements and quotients of definable equivalence relations), so that the forgetful functor $U: \text{Mod}(T') \to \text{Mod}(T)$ yields the right adjoint of a reflection $L \dashv U$. I will say that $L$ is reasonable if it preserves ultraproducts. I will call *a reasonable reflection* the reflections of this kind.


**Prop.** Every reasonable reflection is induced adding formulas of the following form $$\forall \bar x \phi(\bar x) \Rightarrow \exists ! \bar y \psi(\bar y) \wedge \bar y = f(\bar x).$$


**Sketch of proof.** Consider the canonical structure of ultracategory ([à la Makkai](https://www.sciencedirect.com/science/article/pii/000187088790020X?via%3Dihub)) on the categories of models. Recall that by studying their associated pretopos of coordinates, and [because we assumed it has models with at least two elements and quotients of definable equivalence relations](https://mathoverflow.net/questions/418532/why-does-the-category-of-definable-sets-of-t-mathrmeq-have-coproducts/418535#418535), we obtain the following reconstruction result dues to conceptual completeness.
$$ \text{Ult}(\text{Mod}(T), \text{Set}) \simeq \text{Syn}(T).$$


Thus, the reasonable (!) reflection at the level of categories of models, induced a reflection at the level of syntactic categories. Reflections correspond to orthogonality formulas à la Adamek-Rosicky (see Rem 5.6 in Locally presentable and accessible categories).


