The analog of the adjoint functor theorem for indexed categories.
Let be a cartesian category, let and be -indexed categories which are locally small and have all colimits, and suppose further that is well-copowered and has a separating family. Then an indexed functor has an indexed right adjoint precisely iff it is cocontinuous.
This is (Johnstone, theorem B2.4.6).
Section B2.4 in
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