preserves the structures of triangulated categories).

We will say that a class ${\mathcal{P}}\subset {\operatorname{Obj}}{\underline{C}}$ is [*suspended*]{} if ${\mathcal{P}}[1]\subset {\mathcal{P}}$; ${\mathcal{P}}$ is [*cosuspended*]{} if ${\mathcal{P}}[-1]\subset {\mathcal{P}}$.

A class ${\mathcal{P}}\subset {\operatorname{Obj}}{\underline{C}}$ is said to be [*extension-closed*]{} if $0\in {\mathcal{P}}$ and for any distinguished triangle $A\to B\to C$ in ${\underline{C}}$ we have the implication $A,C\in
{\mathcal{P}}\implies B\in {\mathcal{P}}$. In particular, an extension-closed ${\mathcal{P}}$ is [*strict*]{} (i.e., contains all objects of ${\underline{C}}$ isomorphic to its elements). The smallest extension-closed class ${\mathcal{P}}'$ containing a given ${\mathcal{P}}\subset {\operatorname{Obj}}{\underline{C}}$ will be called the [*extension-closure*]{} of ${\mathcal{P}}$. The smallest extension-closed Karoubi-closed ${\mathcal{P}}'\subset {\operatorname{Obj}}{\underline{C}}$ containing ${\mathcal{P}}$ will be called the [*envelope*]{} of ${\mathcal{P}}$.

We call the smallest Karoubi-closed triangulated subcategory ${\underline{D}}$ of ${\underline{C}}$ such that ${\operatorname{Obj}}{\underline{D}}$ contains ${\mathcal{P}}$ the [*triangulated subcategory densely generated by*]{} ${\mathcal{P}}$; we will write ${\underline{D}}={\langle}{\mathcal{P}}{\rangle}_{{\underline{C}}}$.

We will say that ${\underline{C}}$ [*has coproducts*]{} (resp. products, resp. countable coproducts) whenever it contains arbitrary small coproducts (resp. products, resp. countable coproducts) of families of its objects.

${\underline{A}}$ will usually denote some abelian category; the case ${\underline{A}}={\underline{\operatorname{Ab}}}$ is the most important one for the purposes of this paper.

We will call a covariant additive functor ${\underline{C}}\to {\underline{A}}$ for an abelian ${\underline{A}}$ [*homological*]{} if it converts distinguished triangles into long 