Module categories cartesian fibration

Proposition 19.1.16 ([Lurie (2017), Corollary 4.2.3.2]). Let \(D\) be left-tensored over a monoidal \(\infty \)-category \(C\). The restriction functor \[ \fa ^*\colon \LMod (D)\longrightarrow \Alg (C) \] is a cartesian fibration. A morphism in \(\LMod (D)\) is \(\fa ^*\)-cartesian if and only if its underlying morphism in \(D\) is an isomorphism. Consequently, a morphism \(f\colon A\to B\) of associative algebras induces a restriction-of-scalars functor \[ f_*\colon \LMod _B(D)\longrightarrow \LMod _A(D) \] which does not change the underlying object of \(D\).