The analog of the adjoint functor theorem for indexed categories.

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).

- R. Parรฉ, D. Schumacher,
*Abstract families and the adjoint functor theorems*, in*Indexed categories and their applications*, Lecture Notes in Math. vol 661 Springer (1978)

Section B2.4 in

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