Lemma 13.4.12. Let $\mathcal{D}$ be a pre-triangulated category. Let $I$ be a set.

Let $X_ i$, $i \in I$ be a family of objects of $\mathcal{D}$.

If $\prod X_ i$ exists, then $(\prod X_ i)[1] = \prod X_ i[1]$.

If $\bigoplus X_ i$ exists, then $(\bigoplus X_ i)[1] = \bigoplus X_ i[1]$.

Let $X_ i \to Y_ i \to Z_ i \to X_ i[1]$ be a family of distinguished triangles of $\mathcal{D}$.

If $\prod X_ i$, $\prod Y_ i$, $\prod Z_ i$ exist, then $\prod X_ i \to \prod Y_ i \to \prod Z_ i \to \prod X_ i[1]$ is a distinguished triangle.

If $\bigoplus X_ i$, $\bigoplus Y_ i$, $\bigoplus Z_ i$ exist, then $\bigoplus X_ i \to \bigoplus Y_ i \to \bigoplus Z_ i \to \bigoplus X_ i[1]$ is a distinguished triangle.

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