Lemma 13.14.12. Assumptions and notation as in Situation 13.14.1. Let $(X, Y, Z, f, g, h)$ be a distinguished triangle of $\mathcal{D}$. If $X, Y$ compute $RF$ then so does $Z$. Similar for $LF$.

Proof. By Lemma 13.14.6 we know that $RF$ is defined at $Z$ and that $RF$ applied to the triangle produces a distinguished triangle. Consider the morphism of distinguished triangles

$\xymatrix{ (F(X), F(Y), F(Z), F(f), F(g), F(h)) \ar[d] \\ (RF(X), RF(Y), RF(Z), RF(f), RF(g), RF(h)) }$

Two out of three maps are isomorphisms, hence so is the third. $\square$

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