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).
Robert Parรฉ, 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
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