Prismatic cohomology and A_-cohomology with coefficients

For a smooth p-adic formal scheme over the ring of integers of a perfectoid field of mixed characteristic (0, p) containing all p-power roots of unity, we prove that the prismatic cohomology of a locally finite free prismatic crystal is isomorphic to the A_-cohomology of the corresponding relative Breuil-Kisin-Fargues module, which is a certain type of locally finite free A_-module, on the proétale site of the generic fiber. We use a global description of the former in terms of q-Higgs modules via cohomological descent. We also discuss its compatibility with inverse image functors, scalar extensions under Frobenius, and tensor products.