**Proposition 9.13.** For $j = 1,2$ let $(\otimes_j, m_j)$ be a triangulated internal tensor structure on $\mathbb{H}_j$, and let $(\boxtimes_j, m_j)$ denote the triangulated external tensor structure corresponding to it (modulo triangulated equivalence) via the bijection of Proposition 9.6.

Then the constructions of Lemma 8.9 and Lemma 8.10 canonically induce mutually inverse bijections between triangulated internal tensor structures on $R$ (with respect to $(\otimes_1, m_1)$ and $(\otimes_2, m_2)$) and triangulated external tensor structures on $R$ (with respect to $(\boxtimes_1, m_1)$ and $(\boxtimes_2, m_2)$).

*Proof.* One has to show that the constructions of Lemma 8.9 and Lemma 8.10 can be canonically promoted to the triangulated setting. To this end, note that there exists an obvious way to make the natural isomorphisms in the external and internal setting into bi-triangulated natural isomorphisms. Thus one only needs to check that this respects the additional conditions of Definition 9.11 and Definition 9.12. This is easy and left to the interested reader. □

*Remark 9.14.* Note that no additional condition is needed in the triangulated case for the associativity, symmetry and unitarity of a morphism as discussed in the second part of Section 8. Again, the reason is that these are properties rather than additional structures.

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