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Let $\mathcal{T}$ be a triangulated category, and let $\mathcal{A}:=\operatorname{End}(\mathcal{T})$ be the category of triangulated endofunctors of $\mathcal{T}$.

Is $\mathcal{A}$ triangulated?

My suspicion is no, due to the lack of functoriality of mapping cones, but yes if $\mathcal{T}$ possesses an enhancement in the sense of Bondal-Kapranov.

Thanks!

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