Prismatic cohomology and $A_{\inf }$-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 $A_{\inf }$-cohomology of the corresponding relative Breuil-Kisin-Fargues module, which is a certain type of locally finite free ${\mathbb{A}}_{\inf }$-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.