Definition A.13.
Let \(\varphi\colon C \to D\) be a cocontinuous functor between presentable categories. We define its kernel as the collection of morphisms in \(C\) that is inverted by \(\varphi\):
\[\ker(\varphi) \quad := \quad \{f\colon X \to Y \in \Ar(C) \mid \varphi(f)\text{ is an isomorphism}\}.\]