Limit of a sequence of locally presentable categories Let $\dotsc \to \mathcal{C}_2 \xrightarrow{F_1} \mathcal{C}_1 \xrightarrow{F_0}\mathcal{C}_0$ be a sequence of cocontinuous functors between locally presentable categories. Consider the limit $\mathcal{C}=\lim_n \mathcal{C}_n$ inside the $2$-category of categories. It consists of sequences $X_n \in \mathcal{C}_n$ of objects equipped with isomorphisms $F_n(X_{n+1}) \to X_n$.
1) Why is $\mathcal{C}$ again locally presentable? (Of course, it is cocomplete. The only problem is to show that there is a strong generating set of presentable objects.)
2) Can we say anything about the explicit description of equalizers in $\mathcal{C}$? (Of course, they are not computed pointwise since the $F_n$ are not continuous.) Probably it will be some kind of coreflection of the pointwise kernel, but in that case I would like to learn more about that coreflection. I am not only interested in existence results.
I am aware of Bird's thesis on limits in $2$-categories of locally presentable categories, but since this apparently only exists as a poorly scanned copy, it is very hard to find the corresponding results. Actually I would like to prefer a self-contained answer if possible.
 A: This is not a self-contained answer, and it only answers (1).  But in case you are not aware of it, someone should mention the result of Makkai-Pare (in their monograph Accessible categories) that the 2-category of accessible categories and accessible functors has (pseudo) limits.  Left or right adjoints between locally presentable categories are automatically accessible, so the limit of your sequence is accessible, and the 2-category of cocomplete categories and cocontinuous functors also has pseudo limits (with colimits computed pointwise), so your limit is cocomplete; hence it is locally presentable.
A: Here's a proof that if $\kappa$ is an uncountable regular cardinal, then the 2-category $\mathsf{Pres}_\kappa$ of locally $\kappa$-presentable categories are closed in $\mathsf{Cat}$ under PIE limits. Here the 1-morphisms are cocontinuous functors which preserve the $\kappa$-presentable objects, or equivalently have $\kappa$-accessible right adjoints. But since $\mathsf{Pres}_\kappa$ is a non-full sub-2-category, the limits in $\mathsf{Cat}$ do not necessarily continue to be limits in $\mathsf{Cat}$. But from the description below it will be clear that the inclusion $\mathsf{Pres}_\kappa \to \mathsf{Cat}$ reflects inserters, equifiers, and $\kappa$-small products.
Products
It's easy to check that locally $\kappa$-presentable categories are closed under small products. For colimits can be computed pointwise, and a generating set is given by objects whose components are $\kappa$-presentable, and initial except on a $\kappa$-small set of coordinates. If we were working with general $\kappa$-accessible categories, this would be more complicated -- I'm not sure, we might have to raise the degree of accessibility. This does not require $\kappa$ to be uncountable. But note that if the product is of size $\kappa$ or more, then generally it will not actually be a product in the 2-category $\mathsf{Pres}_\kappa$ because a functor all of whose projections preserve $\kappa$-presentable objects will generally not itself preserve $\kappa$-presentable objects.
Inserters
Now consider the inserter $Ins(F,G)$ between functors $F,G: \mathcal{C} \to \mathcal{D}$. Colimits are computed as in $\mathcal{C}$, and it's easy to see that the objects whose underlying $\mathcal{C}$-object is $\kappa$-presentable are $\kappa$-presentable. So it suffices to show that the colimit closure of these objects is all of $Ins(F,G)$ -- by Theorem 2.5.1 in Makkai-Paré, or by Lemma 3.7 in these notes by Mike Shulman and the usual strong-generator characterization of locally $\kappa$-presentable categories. Consider an object $F(c) \overset{\gamma}{\to} G(c)$ of $Ins(F,G)$. Then $c$ has some presentability rank $\lambda$. If $\lambda \leq \kappa$ we are done, otherwise $\lambda = \mu^+$ for some uncountable $\mu$, and $c$ is a $\mu$-sized colimit of $\kappa$-presentable objects (see Adámek and Rosický Remark 1.30). Let $K: I \to \mathcal{C}$ be such a diagram. There are two cases: either $\mu$ is regular, or $\mu$ is singular.
If $\mu$ is regular.
In this case, we can replace $I$ with a $\mu$-sized poset with $\mu$-small colimits and $K$ with a functor that preserves these colimits and takes values in the $\mu$-presentable objects of $\mathcal{C}$. To see this, first replace $I$ with its free completion under $\mu$-small colimits (extending $K$ by $\mu$-small-cocontinuity); then an easy induction allows one to choose a cofinal poset closed under $\mu$-small colimits. Construct a cofinal chain $(i_\alpha)_{\alpha < \mu}$ in $I$ and a natural family of morphisms $\gamma_\alpha: F K i_\alpha \to G K i_\alpha$ as follows. Enumerate the objects of $I$ as $(x_\alpha)_{\alpha < \mu}$. Let $i_\alpha^0 = \sup(\{i_\beta \mid \beta < \alpha \} \cup \{x_\alpha\})$. Because the $Fi$'s are $\mu$-presentable and $I$ is $\mu$-filtered, we may inductively choose objects $i_\alpha^n$ and morphisms $i_\alpha^n \to i_\alpha^{n+1}$ and $\gamma_\alpha^n : F K i_\alpha^n \to G K i_\alpha^{n+1}$ such that the following diagram commutes for all $\beta < \alpha$:
$\require{AMScd}
\begin{CD}
FK i_\beta @>>> FKi_\alpha^0 @>>> \cdots @>>> FKi_\alpha^n @>>> FKi_\alpha @>>> Fc \\
@V{\gamma_\beta}VV @V{\gamma_\alpha^0}VV @VVV @V{\gamma_\alpha^n}VV @V{\gamma_\alpha}VV @V{\gamma}VV \\
GK i_\beta @>>> GKi_\alpha^1 @>>> \cdots @>>> GKi_\alpha^{n+1} @>>> GKi_\alpha @>>> Gc 
\end{CD}$
In the diagram, we have set $i_\alpha = \varinjlim_{n<\omega} i_\alpha^n$ and observed that becuse this colimit is preserved by $K$, $F$, and $G$, we can define $\gamma_\alpha : FKi_\alpha \to GKi_\alpha$ to be the colimit of the $\gamma_\alpha^n$'s, and this is indeed how we define $i_\alpha$ and $\gamma_\alpha$. This is where we need $\kappa$ to be uncountable -- otherwise we cannot take the colimit of this chain when $\mu = \kappa$.
Now it is easy to see that the chain $(i_\alpha)_{\alpha < \mu}$ is cofinal in $I$, and the arrows $(FKi_\alpha \overset{\gamma_\alpha}{\to} GKi_\alpha)_{\alpha < \mu}$ define a functor into $Ins(F,G)$ taking values in the $\mu$-presentable objects whose colimit is $Fc \overset{\gamma}{\to} Gc$. By induction, the $\mu$-presentable objects are in the colimit closure of the $\kappa$-presentable objects (this is where we use that $\mu$ is regular -- otherwise $\mu$-presentability is the same thing as $\lambda$-presentability!). So $Fc \overset{\gamma}{\to} Gc$ is in this colimit closure too.
If $\mu$ is singular
The basic idea is to do what we just did for a cofinal sequence of smaller regular cardinals, and take a colimit. (Interestingly, this argument seems to work for any limit cardinal $\mu$).
Choose a sequence of regular cardinals $(\mu_\alpha)$ satisfying $\kappa \leq \sum_{\beta < \alpha} \mu_\beta < \mu_\alpha < \mu$ with supremum $\mu$. Similarly to the preliminaries before, we can replace $I$ with a $\mu$-sized poset equipped with an exhaustive filtration $I_0 \subset I_1 \subset \dots \subset I$, $\cup_\alpha I_\alpha = I$ satisfying the conditions that $|I_\alpha| = \mu_\alpha$ and $I_\alpha$ is closed under $\mu_\alpha$-small colimits, which are preserved by the inclusion $I_\alpha \to I$. And we may assume that $K$ preserves $\mu$-small colimits and that $Ki$ is $\mu_\alpha$-presentable for $i \in I_\alpha$ . We construct a chain $(i_\alpha)$ with $i_\alpha \in I_\alpha$ and a natural family of morphisms $\gamma_\alpha : Fi_\alpha \to Gi_\alpha$ by taking $i_\alpha^0 = \sup (\cup_{\beta < \alpha} I_\beta)$ and performing an iterative construction as before to choose $i_\alpha$ and $\gamma_\alpha$. It is harmless to assume that $i_\alpha \in I_\alpha$ -- otherwise $i_\alpha$ first appears in $I_{\alpha'}$ for some $\alpha'>\alpha$, and we simply modify our choice of $I_\beta$ for $\alpha \leq \beta < \alpha'$ by adding in $i_\alpha$ and closing under $\mu_\beta$-filtered colimits in $I_{\alpha'}$. As before, we have defined a chain in $Ins(F,G)$ whose colimit is $Fc \overset{\gamma}{\to} Gc$. Moreover for each $\alpha$, the object $Ki_\alpha$ is $\mu_\alpha$-presentable, and $\mu_\alpha$ is a regular cardinal strictly less than $\lambda$, so by the inductive hypothesis, $Fc \overset{\gamma}{\to} Gc$ is in the colimit closure of the $\kappa$-presentable objects as desired.
Equifiers
I think that a similar argument to the inserter case will show that locally $\kappa$-presentable categories are closed under equifiers for uncountable $\kappa$.
