Stability properties of essential geometric morphisms Notation.

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*$\mathsf{Topoi}$ is the bicategory of topoi, geometric morphisms and natural transformations between left adjoints.

*$\mathsf{Topoi}_{\text{ess}}$ is the bicategory of topoi, essential geometric morphisms and natural transformations between left adjoints.

*$\mathsf{Presh}$ is the full subcategory of $\mathsf{Topoi}$ spanned by presheaf topoi.

*$\mathsf{Presh}_{\text{ess}}$ Is the full subcategory of $\mathsf{Topoi}_{\text{ess}}$ spanned by presheaf topoi.

Questions.

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*Is there a reference for the bicategorical properties of $\mathsf{Topoi}_{\text{ess}}$, $\mathsf{Presh}$, $\mathsf{Presh}_{\text{ess}}$?

*Which (pseudo)(co)limits are preserved by the inclusion $\mathsf{Presh}_{\text{ess}} \subset \mathsf{Topoi}$? (This is my motivating question.)

 A: This is only a partial answer. With '(co)limit' I will always mean pseudo(co)limit.

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*If $\mathcal{C}$ and $\mathcal{D}$ are Cauchy-complete, then the category of essential geometric morphisms $\mathbf{PSh}(\mathcal{C}) \to \mathbf{PSh}(\mathcal{D})$ (and geometric transformations between them) is equivalent to the opposite of the category of functors $\mathcal{C} \to \mathcal{D}$ (and natural transformations between them). This is in “Sketches of an Elephant”, Part A, Example 4.1.4 and Lemma 4.1.5. So in this sense $\mathsf{Presh}_\mathrm{ess}$ is a full subcategory of $\mathsf{Cat}^\mathrm{co}$, the bicategory of small categories, functors, and natural transformations (with the natural transformations in the opposite direction).

*By using the above, I think it follows that $$\mathrm{colim}_i\, \mathbf{PSh}(\mathcal{C}_i) ~\simeq~ \mathbf{PSh}(\mathrm{colim}_i \,\mathcal{C}_i)$$ in $\mathsf{Presh}_\mathrm{ess}$, as long as $\mathrm{colim}_i\, \mathcal{C}_i$ is still Cauchy-complete. In particular, coproducts are computed as $\bigsqcup_i \mathbf{PSh}(\mathcal{C}_i) \simeq \mathbf{PSh}(\bigsqcup_i \mathcal{C}_i)$, and this agrees with the coproduct in $\mathsf{Topoi}$. So the inclusion $\mathsf{Presh}_\mathrm{ess} \subset \mathsf{Topoi}$ preserves coproducts, in particular the initial object.   Similarly, I think that $$\mathrm{lim} \, \mathbf{PSh}(\mathcal{C}_i) \simeq \mathbf{PSh}(\mathrm{lim}_i \mathcal{C}_i)$$ in $\mathsf{Presh}_\mathrm{ess}$, as long as $\lim_i \mathcal{C}_i$ is still Cauchy-complete. In particular, the terminal object is $\mathbf{PSh}(1) \simeq \mathbf{Sets}$, just like in $\mathsf{Topoi}$. Also, the product of $\mathbf{PSh}(\mathcal{C})$ and $\mathbf{PSh}(\mathcal{D})$ is $\mathbf{PSh}(\mathcal{C}\times\mathcal{D})$. This is also the product in $\mathsf{Topoi}$ (see Pitts, "On product and change of base for toposes"). I don’t know whether $\mathsf{Presh}_\mathrm{ess} \subset \mathsf{Topoi}$ preserves pullbacks. Examples seem to suggest that pullbacks are preserved, I would be very interested in a proof (here by 'pullback' I mean pseudo-pullback).
Update: here is an example showing that arbitrary products are not preserved. Consider the categories $(\mathcal{C}_n)_{n \in \mathbb{N}}$ with $\mathcal{C_n}$ given by the discrete category on two objects, for each $n \in \mathbb{N}$. Then in $\mathsf{Presh}_\mathrm{ess}$ the product $\prod_{n \in \mathbb{N}} \mathbf{PSh}(\mathcal{C_n})$ is given by $\mathbf{PSh}(\mathcal{D})$, where $\mathcal{D} \simeq \prod_{n \in \mathbb{N}} \mathcal{C}_n$ is the discrete category with $2^{|\mathbb{N}|}$ objects. However, in $\mathsf{Topoi}$ the product is $\prod_{n \in \mathbb{N}} \mathbf{PSh}(\mathcal{C_n}) \simeq \mathbf{Sh}(X)$, where $X$ is the Cantor space (the product in the category of toposes/locales agrees with the product in the category of topological spaces, because it is a countable product of completely metrizable spaces, see Isbell, "Atomless parts of spaces").
Update 2: I believe the inclusion $\mathsf{Presh}_\mathrm{ess} \subset \mathsf{Cat}^\mathrm{co}$ has a left adjoint given by $\mathcal{C} \mapsto \mathbf{PSh}(\mathcal{C})$. This can be used to show that in $\mathsf{Presh}_\mathrm{ess}$ the colimit of $\mathbf{PSh}(\mathcal{C}_i)$'s, with each $\mathcal{C}_i$ Cauchy-complete, is given by $\mathbf{PSh}(\mathcal{D})$ where $\mathcal{D} = \mathrm{colim}\, \mathcal{C}_i$ is the colimit in $\mathsf{Cat}^\mathrm{co}$ (the category $\mathcal{D}$ does not have to be Cauchy-complete in order for this to work).
To show that $\mathbf{PSh}(\mathcal{D})$ is also the colimit in $\mathsf{Topoi}$ (so colimits are preserved), we can use that colimits in $\mathsf{Topoi}$ are computed as the limit in $\mathsf{Cat}$ of the corresponding diagram of inverse image functors (see here). Further, in $\mathsf{Cat}$ we have that $$\mathrm{lim}_i\, \mathbf{PSh}(\mathcal{C}_i) \simeq \mathrm{lim}_i \, \mathrm{Fun}(\mathcal{C}_i^\mathrm{op}, \mathbf{Sets}) \simeq \mathrm{Fun}(\mathrm{colim}_i\, \mathcal{C}_i^\mathrm{op}, \mathbf{Sets}) \\ \simeq \mathrm{Fun}(\mathcal{D}^\mathrm{op}, \mathbf{Sets}) \simeq \mathbf{PSh}(\mathcal{D}).$$
Here we use that $\mathbf{Fun}(-,-)$ sends colimits to limits in the first argument.
The proof above is based on the answer by Yonatan Harpaz here.

