  We characterize the relative prismatic cohomology of Bhatt and Scholze by a
universal property by endowing it with the additional structure of a ``derived
$\delta$-ring". This involves introducing an analogue of prismatic envelopes in
the setting of filtered derived commutative rings and applying this
construction to the Hodge filtration on infinitesimal cohomology. In
particular, we recover relative prismatic cohomology from infinitesimal
cohomology via a purely algebraic process.
