Let $\mathbf C$ and $\mathbf D$ be small categories. $\mathrm{Ind}(\mathbf C)$ is an accessible category (by definition), and is locally finitely presentable (i.e. cocomplete, or equivalently complete) iff $\mathbf C$ has finite colimits. Let $\mathbf C$ and $\mathbf D$ have finite colimits, and consider a functor $F : \mathbf C \to \mathrm{Ind}(\mathbf D)$. By the universal property of $\mathrm{Ind}$, this extends to a finitary functor $\tilde F : \mathrm{Ind}(\mathbf C) \to \mathrm{Ind}(\mathbf D)$.


In terms of $F$, assuming such a characterisation exists:


1. When is $\tilde F$ continuous? (Equivalently, when does $\tilde F$ have a left adjoint?)
2. When is $\tilde F$ cocontinuous? (Equivalently, when does $\tilde F$ have a right adjoint?)


I imagine (2) should hold when $F$ preserves finite colimits, though I wasn't able to find a reference in *Locally presentable and accessible categories*.



When is a finitary functor induced by Ind (co)continuous
Allow me to generalise to $\kappa$-accessible categories for infinite regular cardinals $\kappa$. Your guess for (2) is correct: if $F$ preserves $\kappa$-small colimits then $\tilde{F}$ preserves colimits. The proof is a little bit indirect.


**Proposition.** Let $\mathcal{I}$ be a category and let $\mathcal{C}$ be a small category with $\kappa$-small colimits. If $\mathcal{I}$ is $\kappa$-small, then the comparison functor
$$\textbf{Ind}\_\kappa ([\mathcal{I}, \mathcal{C}]) \to [\mathcal{I}, \textbf{Ind}\_\kappa (\mathcal{C})]$$
is fully faithful and essentially surjective on objects.


(The key point is to show that every diagram of shape $\mathcal{I}$ can be written as a $\kappa$-filtered colimit of diagrams of $\kappa$-presentable objects of the same shape $\mathcal{I}$. Actually, the statement of the proposition is equivalent to this fact, and this is what we need for the next step.)


**Proposition.** Let $\mathcal{C}$ be a category with $\kappa$-small colimits, let $\mathcal{E}$ be a category with $\kappa$-filtered colimits, let $F : \mathcal{C} \to \mathcal{E}$ be a functor, and let $\tilde{F} : \textbf{Ind}\_\kappa (\mathcal{C}) \to \mathcal{E}$ be the extension. Then $\tilde{F}$ preserves colimits if and only if $F$ preserves $\kappa$-small colimits.


(We already know $\tilde{F}$ preserves $\kappa$-filtered colimits, so it is enough to check whether $\tilde{F}$ preserves $\kappa$-small colimits. The "only if" direction is easy. The "if" direction is proved using the previously mentioned decomposition of $\kappa$-small diagrams.)


I don't have a good answer for (1), but perhaps this will be enough for your purposes:


**Proposition.** Let $\mathcal{C}$ and $\mathcal{D}$ be small categories with $\kappa$-small colimits. A functor $F : \mathcal{C} \to \mathcal{D}$ has a left adjoint if and only if $\textbf{Ind}\_\kappa (F) : \textbf{Ind}\_\kappa (\mathcal{C}) \to \textbf{Ind}\_\kappa (\mathcal{D})$ has a left adjoint.


(The "only if" direction is easy: after all, $\textbf{Ind}\_\kappa$ is a pseudofunctor so it preserves adjunctions. The "if" direction amounts to saying that the left adjoint of a $\kappa$-accessible functor between locally $\kappa$-presentable categories preserves $\kappa$-presentable objects, which is straightforward to check.)


