Consider the category $\mathbf{Top}$ of topological spaces, the category $\mathbf{Topos}$ of toposes and geometric morphisms, and the category $\mathbf{Loc}$ of locales. Let
$$\mathrm{Sh}\colon\mathbf{Top}\to\mathbf{Topos}$$
be the functor sending a space $X$ to the topos of sheaves on $X$. Does this functor have a left or a right adjoint?


Of course, $\mathrm{Sh}$ factorizes as
$$\mathbf{Top}\to\mathbf{Loc}\to\mathbf{Topos},$$
since the sheaves on $X$ are the sheaves on the locale of open subsets of $X$.


It is well-known that $\mathbf{Top}\to\mathbf{Loc}$ has a right adjoint and $\mathbf{Loc}\to\mathbf{Topos}$ has a left adjoint ($\mathbf{Loc}$ is reflective in $\mathbf{Topos}$). So the question whether $\mathrm{Sh}$ has a left adjoint reduces to the question whether $\mathbf{Top}\to\mathbf{Loc}$ has a left adjoint and the question whether $\mathrm{Sh}$ has a right adjoint reduces to the question whether $\mathbf{Loc}\to\mathbf{Topos}$ has a right adjoint.


*Remark.* I already asked this on math.SE but I didn't get an answer.



Does the functor $\mathrm{Sh}\colon\mathbf{Top}\to\mathbf{Topos}$ have an adjoint?
In this answer, Topos is interpreted as a 2-category.
(As a side remark, the 1-category of toposes does not make sense
until one picks a specific model for toposes and geometric morphisms, and different models
need not be equivalent as 1-categories.
For the 1-categorical framework to make sense, at the very least
one needs to organize toposes into a relative category,
so that different models can be shown to be Dwyer–Kan equivalent as relative categories.)



> 
> whether Top→Loc has a left adjoint
> 
> 
> 


The functor Top→Loc does not have a left adjoint because it does not preserve finite products.
For example, the product of rational numbers with themselves
as locales and as topological spaces produces nonisomorphic locales.
In particular, rational numbers form a topological group, but not a localic group.



> 
> whether Loc→Topos has a right adjoint
> 
> 
> 


Loc→Topos does not have a right adjoint because it does not preserve homotopy colimits.


For example, suppose G is a discrete group acting on a point.
Then the colimit of this action in locales is again a point.
But the homotopy colimit of this action in toposes
is the delooping of G, which is not equivalent to a point.


Thus, both Loc→Topos and Top→Topos do not have a right adjoint functor.


