By the general properties of adjoint (∞,1)-functors it is sufficient to show that . To see this, we use that every ∞-groupoid ∞Grpd is the (∞,1)-colimit (as discussed there) over itself of the (∞,1)-functor constant on the point: .
The left adjoint preserves all (∞,1)-colimits, but if has a right adjoint, then it does, too, so that for all we have
Now , being a right adjoint preserves the terminal object and so does by definition of (∞,1)-geometric morphism. Therefore
We check that the global section (∞,1)-geometric morphism ∞Grpd preserves (∞,1)-colimits.
The functor is given by the hom-functor out of the terminal object of , this is :
The hom-∞-groupoids in the over-(∞,1)-category are (as discussed there) homotopy fibers of the hom-sapces in : we have an (∞,1)-pullback diagram
Overserve that (∞,1)-colimits in the over-(∞,1)-category are computed in .
If is small-projective then by definition we have
Inserting all this into the above -pullback gives the -pullback
By universal colimits in the (∞,1)-topos ∞Grpd, this (∞,1)-pullback of an (∞,1)-colimit is the -colimit of the separate pullbacks, so that
So does commute with colimits if is small-projective. Since all (∞,1)-toposes are locally presentable (∞,1)-categories it follows by the adjoint (∞,1)-functor that has a right adjoint (∞,1)-functor.