Let *C, E* be small categories, let *Ĉ* = Set*C*op, and let *F:Ĉ → Ê* be cocontinuous. I think *F* will always have a right adjoint when *C, E* are small, but not necessarily if they're large. Is that right?



Does every cocontinuous functor between categories of presheaves on small categories have a right adjoint?
The part "*F* will always have a right adjoint when *C*, *E* are small" is definitely right. Using some mildly overkill machinery: in this case *Ĉ* and *Ê* are locally presentable categories, and the result is then the **adjoint functor theorem for locally presentable categories**:


Theorem: Let C and D be locally presentable categories and F : C → D a functor. Then


1. F has a right adjoint iff F preserves small colimits.
2. F has a left adjoint iff F is accessible (preserves κ-filtered colimits for some κ) and preserves small limits.


(Reference: *Higher Topos Theory* Corollary 5.5.2.9 for the (∞,1)-categorical version)


