Pointed ∞-categories and reduced functors

Notation 4.1.11. Let \(\Cat _{\infty }^{\term } \subseteq \Cat _{\infty }\) denote the subcategory of \(\infty \)-categories with a terminal object, and functors preserving the terminal object. Let \(\Cat _{\infty }^{\pt } \subseteq \Cat _{\infty }^{\term }\) denote the full subcategory of pointed \(\infty \)-categories. Given \(\infty \)-categories \(C\) and \(D\) with terminal objects, we denote by \[ \Fun _*(C,D) \quad \subseteq \quad \Fun (C,D) \] the full subcategory spanned by functors preserving the terminal object.