Let $\sigma$ and $\tau$ be first-order languages and let $S$ resp. $T$ be theories (sets of sentences) for $\sigma$ resp. for $\tau$ ($S$ and $T$ possibly empty). Let $\mathcal C$ resp. $\mathcal D$ be full subcategories of the categories of models for $S$ resp. $T$, that is, objects are models and maps are elementary embeddings. Let $ F: \mathcal C \to \mathcal D$ be a functor.

Question: What are necessary and sufficient conditions for the image $F(Ob(\mathcal C))$ to be axiomatizable within $\mathcal D$ with respect to $\tau$? In other words, when is there a theory $\Sigma$ for $\tau$ such that $F(Ob(\mathcal C)) = \{O \in Ob(\mathcal D) \mid O \models \Sigma \}$.

For instance, the class of additive groups of rings is not axiomatizable, see this post. Equivalently, the image of the forgetful functor from rings to abelian groups cannot be axiomatized within the language of groups. The same is true for multiplicative monoids of rings.

It seems that if the functor does not preserve enough information of the symbols in $\sigma$, it is not possible to axiomatize its image. Can this be made more concrete?

Thank you in advance for any help!