The $2$-category of topoi and geometric morphisms is not locally small. For example, if $A$ is the classifying topos for abelian groups, the category of geometric morphisms from $Set$ to $A$ is equivalent to the category of all abelian groups, which is not small. This problem of course persists for higher topoi, since ordinary topoi embedd into them.


**Question:** Suppose that $\mathcal{E}$ and $\mathcal{F}$ are two topoi (or higher topoi) such that I know that $Hom\left(\mathcal{E},\mathcal{F}\right)$ is small. It is easy to show that for any $F$ in $\mathcal{F},$ $Hom\left(\mathcal{E},\mathcal{F}/F\right)$ is also small. Can it also be shown that for all $E$ in $\mathcal{E},$ $Hom\left(\mathcal{E}/E,\mathcal{F}\right)$ is small?



Local smallness and (higher) topoi
Consider a theory which has no models in $\mathrm{Set}$, but has a model in $\mathrm{Sh}(L)$ for some locale $L$. For example, the theory $\mathcal{CLF}$ of complete linearly ordered fields with more than $\sharp \mathbb{R}$ number of elements will do. For $ L $ we can take Barr or Diaconescu covering of $\mathbb {B}\mathcal {CLF} $. Take some non-small theory, like abelian groups $\mathcal{A}b$, and consider a classifying topos $\mathrm {B} T$ for pairs $T = (K: \mathcal{CLF}, A: \mathcal{A}b)$. For the topos $\mathcal E $ consider $$\mathrm {Sh} (L + pt) = \mathrm {Sh}(L) + \mathrm {Set} $$
An etale map $ L \to L + pt $ corresponds to some sheaf $ E \in \mathcal E $, and $\mathcal {E}/E = \mathrm {Sh}(L) $. 


$ T$ will have no models in $\mathcal{E}$, but a non-small category of models in $\mathcal{E}/E$.


*Added later*:


Instead of $\mathcal{CLF}$ we could take any theory without $\mathrm{Set}$-models. For example we could take a geometric theory, corresponding to some locale without points (just in case if there happen to be some obscure smallness problems with the following statements).


For any topos $\mathcal{F}$ we have $$Mod\_T (\mathcal{F}) = Mod\_{\mathcal{CLF}}(\mathcal{F}) \times Mod\_{\mathcal{A}b} (\mathcal{F})$$
since a model of $T$ is just a pair of models ($Mod\_T(\mathcal{F})$ is the category of T-models in $\mathcal{F}$). Since $L$ is the Barr covering for $\mathcal{CLF}$, $Mod\_{\mathcal{CLF}}(\mathrm{Sh}(L))$ is a non-empty category. Thus to prove that $Mod\_T(\mathcal{F})$ is non-small, we only need to prove it for $Mod\_{\mathcal{A}b}(\mathcal{F})$. The category $Mod\_{\mathcal{A}b}(\mathrm{Set})$ is clearly non-small. I claim that the category of constant $\mathcal{A}b$-valued sheaves on $\mathrm{Sh}(L)$ is a non-small subcategory of $Mod\_{\mathcal{A}b}(\mathcal{F})$.


The terminal morphism $p\colon L \to pt$ gives an adjoint pair of functors $p\_\* \colon \mathrm{Sh}(L) \leftrightharpoons \mathrm{Set} \colon p^\*$, $p^\* \vdash p\_\*$. Sheaves on locale $L$ correspond to etale spaces over $L$, and the pullback functor $p^\*$ is simply the pullback of corresponding etale spaces, i.e. it maps $S\in \mathrm{Set}$ to $\left( L\otimes S \to L \right) \in Et(L)$, the morphism to $L$ being the obvious "collapsing fibers" projection $L\otimes S \to L \otimes pt = L$. Here $L \otimes S$ stands for a coproduct of $S$ copies of $L$ in category of locales. Now the statement that $p^\*: \mathcal{A}b(\mathrm{Set}) \to \mathcal{A}b(\mathrm{Sh}(L))$ is injective looks like something that should be obviously true (for $A\in \mathcal{A}b(\mathrm{Set})$ the sheaf of abelian groups $p^\*(A)$ should have global sections roughly like $A^{\pi\_0(L)}$), but at the moment I can only formalize the proof in the following roundabout way.


An isomorphism $p^\*(A) \simeq p^\*(B)$, $A,B \in \mathcal{A}b(\mathrm{Set})$ would give an isomorphism of etale spaces $L\otimes A \simeq L\otimes B$. Since $\mathcal{L}oc = \mathcal{F}rm^{op}$, this would give an isomorphism $\mathcal{O}(L)^A \simeq \mathcal{O}(L)^B$, $\mathcal{O}\colon \mathcal{L}oc^{op} \simeq \mathcal{F}rm$. The product in $\mathcal{F}rm$ is induced from the product in $\mathrm{Set}$, like in any algebraic theory. Thus if $|A| > |\mathcal{O}(L)|$ and $|B| > |\mathcal{O}(L)|$, then $\mathcal{O}(L)^A$ and $\mathcal{O}(L)^B$ are non-isomorphic even as sets, moreso as frames. Here $|A|$ stands for cardinality. Since we have a non-small set of groups with cardinality greater than $|\mathcal{O}(L)|$ (e.g. all big enough free groups), the statement is proved.


