Show that a functor which preserves colimits has a right adjoint

Question

In Moerdijk, Classifying spaces and classifying topoi, page 22, we find the following statement: a functor between topoi which preserves colimits must have a right adjoint, necessarily unique up to isomorphism (MacLane, Categories for the Working Mathematician, page 83).

Despite the reference, I actually fail to see the motivation for this. Can someone give me a hint for a proof (or an indication on how the reference is related to the problem)? It seems to me that the Theorem in CWM only covers the "uniqueness" part...

Thank you in advance.

Answer 1

I don't know a reference for elementary topoi, but one for presheaf topoi can be found around pages 41-43 of MacLane-Moerdijk Sheaves in Geometry and Logic: they prove (Corollary 4) that a functor $A:\mathbb{C}\rightarrow\mathcal{E}$ where $\mathcal{E}$ is cocomplete and $\mathbb{C}$ is small extends along the Yoneda embedding $y$ to a unique (up to iso) $L_A:\widehat{\mathbb{C}}\rightarrow\mathcal{E}$ which preserves colimits, and such a functor has a right adjoint by construction (Theorem 2).

If now you consider some colimit preserving functor $F:\widehat{\mathbb{C}}\rightarrow\widehat{\mathbb{D}}$, the composition $A=Fy$ must extend in such a way: so it extends to $L_A=F$, but then $F$ must have a right adjoint.

Answer 2

This is false in general without further hypotheses; see adjoint functor theorem. The reference to CWM, as you say, is only a reference for the uniqueness. The various adjoint functor theorems do imply this statement for a functor between Grothendieck topoi.