  We study relative Wach modules generalising our previous works on this
subject. Our main result shows a categorical equivalence between relative
integral Wach modules and lattices inside relative crystalline representations.
Using this result, we deduce a purity statement for relative crystalline
representations and provide a criteria for checking crystallinity of relative
$p$-adic representations. Furthermore, we interpret relative Wach modules as
modules with $q$-connections. Moreover, for a crystalline representation, we
show that its associated Wach module (together with the Nygaard filtration) is
the canonical $q$-deformation of the filtered $(\varphi,\partial)$-module
associated to the representation.
