A functor
is a $\kappa$-accessible functor (for $\kappa$ a regular cardinal) if $C$ and $D$ are both $\kappa$-accessible categories and $F$ preserves $\kappa$-filtered colimits. $F$ is an accessible functor if it is $\kappa$-accessible for some regular cardinal $\kappa$.
It is immediate from the definition that accessible functors are closed under composition.
If $\lambda\le\kappa$, then every $\kappa$-filtered colimit is also $\lambda$-filtered, and thus if $F$ preserves $\lambda$-filtered colimits then it also preserves $\kappa$-filtered ones. Therefore, if $F$ is $\lambda$-accessible and $C$ and $D$ are $\kappa$-accessible, then $F$ is $\kappa$-accessible. Two conditions under which this happens are:
$C$ and $D$ are locally presentable categories.
$\lambda$ is sharply smaller than $\kappa$, i.e. $\lambda\lhd\kappa$.
In particular, for any accessible functor $F$ there are arbitrarily large cardinals $\kappa$ such that $F$ is $\kappa$-accessible, and if the domain and codomain of $F$ are locally presentable then $F$ is $\kappa$-accessible for all sufficiently large $\kappa$.
For any accessible functor $F$, there are arbitrarily large cardinals $\kappa$ such that $F$ is $\kappa$-accessible and preserves $\kappa$-presentable objects. Indeed, this can be achieved simultaneously for any set of accessible functors. See Adamek-Rosicky, Theorem 2.19.
Assuming the existence of a proper class of strongly compact cardinals, the following are equivalent for the essential image $K$ of an accessible functor:
Assuming the existence of a proper class of strongly compact cardinals, the closure of the image of an accessible functor under passage to subobjects is an accessible subcategory.
The existence of a proper class of strongly compact cardinals can be weakened, see the paper of Brooke-Taylor and Rosický.
Given locally presentable categories $C$ and $D$ and a functor $F\colon C\to D$, if $F$ has a left or right adjoint, then it is an accessible functor.
By Example it follows that polynomial endofunctors of $Set$ are accessible, as they are composites of adjoint functors.
The theory of accessible 1-categories is described in
Essential images of accessible functors are considered in
An improvement of Rosický’s result is in
The theory of accessible $(\infty,1)$-categories is the topic of section 5.4 of
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