Corollary 22.2.4 (Presentability of kernels). Let \(F\colon C\to D\) be a colimit-preserving functor between pointed presentable \(\infty \)-categories. Then the full subcategory \[ \ker (F):=\{X\in C\mid F(X)\simeq 0\}\subseteq C \] is presentable, and its inclusion into \(C\) preserves small colimits.
Proof. The functor \(*\to D\) selecting the zero object is a left adjoint, since the zero object is initial. The pullback
may therefore be formed in \(\PrL \). By Theorem 22.2.3, its underlying \(\infty \)-category is the pullback in \(\widehat {\Cat }_{\infty }\), which identifies it with the displayed full subcategory. The projection \(\ker (F)\to C\) is a morphism in \(\PrL \), so it preserves small colimits. □
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