Definition 4.2.24 (Left exact \(\infty \)-categories). An \(\infty \)-category \(C\) is called left exact, or said to admit finite limits, if it admits all \(I\)-indexed limits for all finite \(\infty \)-categories \(I\). Similarly, a functor \(F\colon C \to D\) between left exact \(\infty \)-categories is called left exact if it preserves \(I\)-indexed limits for all \(I \in \Cat ^{\fin }_{\infty }\). We denote by \[ \Cat ^{\lex }_{\infty } \quad \subseteq \quad \Cat _{\infty } \] the subcategory consisting of the left exact \(\infty \)-categories and left exact functors. Given two left exact \(\infty \)-categories \(C\) and \(D\), we denote by \[ \Fun ^{\lex }(C,D) \quad \subseteq \quad \Fun (C,D) \] the full subcategory spanned by the left exact functors.
One dually obtains a notion of right exactness in terms of finite colimits.
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