My question is ultimately about the model theory of $L_{\infty, \omega}$, but it is more convenient to phrase it in terms of category theory. Suppose I have finitely accessible categories $\mathcal{C}, \mathcal{D}, \mathcal{E}$ and functors $F : \mathcal{C} \to \mathcal{E}$ and $G : \mathcal{D} \to \mathcal{E}$ that preserve filtered colimits and finite-presentability of objects.
Suppose $G : \mathcal{D} \to \mathcal{E}$ is a fully faithful isofibration. Then we may assume without loss of generality that $\mathcal{D}$ is a full subcategory of $\mathcal{E}$ closed under isomorphisms and $G$ is the inclusion. Let $F^{-1} \mathcal{D}$ be the preimage of $\mathcal{D}$ in $\mathcal{C}$; note that this is a full subcategory closed under isomorphisms.
- Is $F^{-1} \mathcal{D}$ necessarily a finitely accessible category?
It is certainly $\aleph_1$-accessible – but the only proof I know of uses a kind of back-and-forth construction which leaves the realm of finitely presentable objects.
- If so, does the inclusion $F^{-1} \mathcal{D}$ preserve the finite-presentability of objects?
In a closely related situation, the claim is false: take $\mathcal{C} = [\mathbb{2}, \mathbf{Set}]$, $\mathcal{D} = \mathbf{Set}$, $\mathcal{E} = \mathbf{Set} \times \mathbf{Set}$, $F$ the forgetful functor and $G$ the diagonal inclusion; then $F^{-1} \mathcal{D}$ is isomorphic to the category of sets equipped with an endomorphism, hence finitely accessible; but $(\mathbb{N}, s)$ is finitely presentable (indeed, free on one generator) in $F^{-1} \mathcal{D}$ but not finitely presentable as an object in $\mathcal{C}$. It's worth noting that $F$ is an isofibration in this (non)example, so the strict pullback still computes the bicategorical pullback.
If the answer to both questions is yes, then that would yield an easy proof of Theorem A of this preprint of Raptis and Rosický. As such it seems the proof will be quite hard, if there is one. However, I am more interested in the speical case where we assume $F : \mathcal{C} \to \mathcal{D}$ is faithful and injective on objects.