Corollary 9.1.14 of ibid. \[ibou2\]. The “calculation” of $g$ is given by Proposition 9.1.8 of ibid. also. It remains to apply Theorem 9.1.16 of ibid.

\[ibou3\]. Corollary 9.1.14 of ibid. allows to deduce the assertion from the previous one. IV. The “only if” part of the assertion is given by assertion III.\[ibou1\]. The converse implication is given by Proposition 9.1.18 of ibid.

Now we recall some terminology, notation, and statements related to infinite coproducts and products in triangulated categories. Some of these definitions and results may will be generalized in §\[scomp\] below.

All the coproducts and products in this paper will be small. We will say that a subclass ${\mathcal{P}}$ of ${\operatorname{Obj}}{\underline{C}}$ is [*coproductive*]{} (resp. [*productive*]{}) if it is closed with respect to all (small) coproducts (resp. products) that exist in ${\underline{C}}$. For triangulated categories closed with respect to (all) coproducts or products we will just say that these categories have coproducts (resp. products).[^16]

We recall the following very useful statement.

\[pcoprtriang\] Assume that ${\underline{C}}$ has coproducts (resp. products, resp. countable coproducts). Then ${\underline{C}}$ is Karoubian and all (small) coproducts (resp. products, resp. countable coproducts) of distinguished triangles in ${\underline{C}}$ are distinguished.

The first of the assertions is given by 