Definition 23.6.1 (Twisted arrow category). Let \(C\) be an \(\infty \)-category. We define its twisted arrow category \(\Tw (C)\) as the source of the left fibration whose cocartesian straightening is the Hom-functor \(\Hom _C\colon C\catop \times C \to \An \): \[ \Tw (C) := \Un ^{\cc }(\Hom _C\colon C\catop \times C \to \An ). \] In particular, it comes equipped with a left fibration \((s,t)\colon \Tw (C) \to C\catop \times C\) whose fibers are the hom animae of \(C\):
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