ScalingStacks

Proof. Clearly, $t$ is fully faithful when restricted to $\mathcal{C}^{w=0}.$ Since $\mathcal{C}^{w=0}$ is tilting, $t$ is also fully faithful when restricted to $\bigcup_{n}\mathcal{C}^{w=n}.$ Now $\mathcal{C}^{w=0}$ generates $\mathcal{C}$ as triangulated subcategory since $w$ is bounded, see [Bon10, Corollary 1.5.7]. Hence $t$ is fully faithful on $\mathcal{C}$ by induction (dévissage) using the long exact sequence of $\operatorname{Hom}$-groups for distinguished triangles and the $5$-lemma. Essential surjectivity follows from dévissage as well since $\mathcal{C}^{w=0}$ generates $\operatorname{K}^{b}(\mathcal{C}^{w=0})$ as triangulated subcategory. ∎