nLab indexed adjoint functor theorem

Idea

The analog of the adjoint functor theorem for indexed categories.

Statement

Theorem

Let $\mathcal{S}$ be a cartesian category, let $\mathbb{C}$ and $\mathbb{D}$ be $\mathcal{S}$-indexed categories which are locally small and have all colimits, and suppose further that $\mathbb{C}$ is well-copowered and has a separating family. Then an indexed functor $F: \mathbb{C} \to \mathbb{D}$ has an indexed right adjoint precisely iff it is cocontinuous.

This is (Johnstone, theorem B2.4.6).

References

• Robert Paré, Dietmar Schumacher, Abstract families and the adjoint functor theorems, in Indexed categories and their applications, Lecture Notes in Math. 661 Springer (1978) [doi:10.1007/BFb0061361]

• Peter Johnstone, Section B2.4 in: Sketches of an Elephant