$C$.

Given a category $C$ and $X,Y\in{\operatorname{Obj}}C$ we will write $C(X,Y)$ for the set of morphisms from $X$ to $Y$ in $C$. We will say that $X$ is a [*retract*]{} of $Y$ if ${\operatorname{id}}_X$ can be factored through $Y$.[^15] 

For a category $C$ the symbol $C^{op}$ will denote its opposite category.

For a subcategory $D\subset C$ we will say that $D$ is [*Karoubi-closed*]{} in $C$ if it contains all retracts of its objects in $C$. We call the smallest Karoubi-closed subcategory $\operatorname{\operatorname{Kar}}_C(D)$ of $C$ containing $D$ the [*Karoubi-closure*]{} of $D$ in $C$. The [*Karoubi envelope*]{} $\operatorname{\operatorname{Kar}}({\underline{B}})$ (no lower index) of an additive category ${\underline{B}}$ is the category of “formal images” of idempotents in ${\underline{B}}$ (so ${\underline{B}}$ is embedded into an idempotent complete category). ${\underline{\operatorname{Ab}}}$ is the category of abelian groups.

${\underline{C}}$, ${\underline{C}}'$, ${\underline{C}_0}$, ${\underline{D}}$, ${{\underline{D}}_0}$, and ${\underline{E}}$ will always denote certain triangulated categories. ${\underline{C}}$ will often be endowed with a weight structure $w$; we always assume that this is the case in those formulations where $w$ is mentioned without any explanations. For $f\in{\underline{C}}(X,Y)$, where $X,Y\in{\operatorname{Obj}}{\underline{C}}$, we call the third vertex of (any) distinguished triangle $X\stackrel{f}{\to}Y\to Z$ a cone of $f$.

We will often consider some representable and corepresentable functors and 