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?

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.

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.

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?

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 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.

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?

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.