Adequate triple all morphisms

Example 13.1.4. If \(C\) admits all pullbacks, we have the adequate triple \((C, C, C)\) where we take all maps to be both forward and backward maps. This gives rise to a fully faithful embedding \[ \Cat _{\infty }^{\pb } \hookrightarrow \AdTrip , \] where \(\Cat _{\infty }^{\pb } \subseteq \Cat _{\infty }\) is the subcategory spanned by those \(\infty \)-categories which admit pullbacks, and those functors which preserve pullbacks.