Lemma 6.3.13. Let \(F\colon C \to D\) be a t-exact functor between two stable \(\infty \)-categories equipped with t-structures. Then for all \(X \in C\) and \(n \in \Z \), there are natural equivalences \[ \tau _{\leq n} F(X) \iso F(\tau _{\leq n} X) \qquadtext { and } F(\tau _{\geq n} X) \iso \tau _{\geq n} F(X). \]
Proof. We prove the first equivalence; the second is dual. Since \(F\) is t-exact, applying \(F\) to the exact sequence \(\tau _{\geq n+1} X \to X \to \tau _{\leq n} X\) gives another exact sequence \[ F(\tau _{\geq n+1} X) \to F(X) \to F(\tau _{\leq n} X) \] where the first object lies in \(D_{\geq n+1}\) and the last one in \(D_{\leq n}\). It follows that the second map induces an isomorphism \(\tau _{\leq n} F(X) \iso F(\tau _{\leq n} X)\). □
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