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.

1. 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.

2. 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.

It seems likely that the general case is very hard or even false, but I am hoping for a positive answer in the special case where:

• $F : \mathcal{C} \to \mathcal{E}$ is faithful and conservative,
• $\mathcal{C}$, $\mathcal{D}$, and $\mathcal{E}$ have only countably many isomorphism classes of finitely presentable objects, and
• there are only have finitely many morphisms between finitely presentable objects in $\mathcal{C}$, $\mathcal{D}$, and $\mathcal{E}$.

These allow us to make a model-theoretic interpretation of the setup: $\mathcal{E}$ is some category of models of a (many-sorted) theory $\mathbb{T}$ in $L_{\omega_1, \omega}$ (but not an arbitrary one), $\mathcal{D}$ is the category of models of an extension of $\mathbb{T}$ (in the same language), and $\mathcal{C}$ is the category of models of an extension of $\mathbb{T}$ (in a possibly expanded language).

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.

1. 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.

2. 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.

It seems likely that the general case is very hard or even false, but I am hoping for a positive answer in the special case where:

• $F : \mathcal{C} \to \mathcal{E}$ is faithful and conservative,
• $\mathcal{C}$, $\mathcal{D}$, and $\mathcal{E}$ have only countably many isomorphism classes of finitely presentable objects, and
• there are only have finitely many morphisms between finitely presentable objects in $\mathcal{C}$, $\mathcal{D}$, and $\mathcal{E}$.

These allow us to make a model-theoretic interpretation of the setup: $\mathcal{E}$ is some category of models of a (many-sorted) theory $\mathbb{T}$ in $L_{\omega_1, \omega}$ (but not an arbitrary one), $\mathcal{D}$ is the category of models of an extension of $\mathbb{T}$ (in the same language), and $\mathcal{C}$ is the category of models of an extension of $\mathbb{T}$ (in a possibly expanded language).

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.

1. 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.

2. 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.

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.

1. 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.

2. 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.

It seems likely that the general case is very hard or even false, but I am hoping for a positive answer in the special case where:

• $F : \mathcal{C} \to \mathcal{E}$ is faithful and conservative,
• $\mathcal{C}$, $\mathcal{D}$, and $\mathcal{E}$ have only countably many isomorphism classes of finitely presentable objects, and
• there are only have finitely many morphisms between finitely presentable objects in $\mathcal{C}$, $\mathcal{D}$, and $\mathcal{E}$.

These allow us to make a model-theoretic interpretation of the setup: $\mathcal{E}$ is some category of models of a (many-sorted) theory $\mathbb{T}$ in $L_{\omega_1, \omega}$ (but not an arbitrary one), $\mathcal{D}$ is the category of models of an extension of $\mathbb{T}$ (in the same language), and $\mathcal{C}$ is the category of models of an extension of $\mathbb{T}$ (in a possibly expanded language).