Definition 4.1.1. ([Lur17, Def. 4.2.1.28]).
Let \(\mathbb V\) be a (possibly large) monoidal \(\infty\)-category, and \(\mathcal C\) a left \(\mathbb V\)-module \(\infty\)-category. A morphism object between objects \(x,y \in \mathcal C\) is an object \(\underline{\mathrm{Hom}}_{\mathcal C}(x,y) \in \mathbb V\) representing the presheaf \(\mathrm{Hom}_{\mathcal C}(- \otimes x, y) \colon \mathbb V^{\mathrm{op}} \rightarrow\mathcal S\), i.e. equipped with isomorphisms natural in \(v\in \mathbb V\) \[\mathrm{Hom}_{\mathbb V}(v, \underline{\mathrm{Hom}}_{\mathcal C}(x,y)) \simeq \mathrm{Hom}_{\mathcal C}(v \otimes x, y).\] A \(\mathbb V\)-module category \(\mathcal C\) is closed if a morphism object exists between every pair of objects \(x, y \in \mathcal C\). A closed monoidal \(\infty\)-category is a monoidal \(\infty\)-category whose left action on itself is closed.