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.