exact sequences; homological functors ${\underline{C}}^{op}\to {\underline{A}}$ will be called [*cohomological*]{} when considered as contravariant functors from ${\underline{C}}$ into $ {\underline{A}}$.

For additive categories $C,D$ the symbol $\operatorname{\operatorname{AddFun}}(C,D)$ will denote the (possibly, big) category of additive functors from $C$ to $D$. Certainly, if $D$ is abelian then an $\operatorname{\operatorname{AddFun}}(C,D)$-complex $X\to Y\to Z$ is exact in $Y$ whenever $X(P)\to Y(P)\to Y(P)$ is exact (in $D$) for any $P\in {\operatorname{Obj}}C$.

Below we will sometimes need some properties of the Bousfield localization setting (cf. §9.1 of [@neebook]; most of these statements are contained in Propositions 1.5 and 1.6 of [@bondkaprserr]).

\[pbouloc\]

Let $F:{\underline{E}}\to {\underline{C}}$ be a full embedding of triangulated categories; assume that ${\underline{E}}$ is Karoubi-closed in ${\underline{C}}$. Denote by ${\underline{D}}$ the full subcategory of ${\underline{C}}$ whose object class is ${\operatorname{Obj}}{\underline{E}}{{}^{\perp}}$; denote the embedding ${\underline{D}}\to {\underline{C}}$ by $i$.

Then the following statements are valid.

I. A (left or right) adjoint to an exact functor is exact.

II\. ${\underline{D}}$ is a Karoubi-closed triangulated subcategory of ${\underline{C}}$.

III\. Assume that $F$ possesses a right adjoint $G$.

1.  \[ibou1\] Then for any $N\in {\operatorname{Obj}}{\underline{C}}$ there exists a distinguished triangle $$\label{ebou}
    N'\stackrel{f}{\to} N\stackrel{g}{\to} N''\to N'[1]$$ with $N'\in   {\operatorname{Obj}}{\underline{E}}$ and $ N'' \in{\operatorname{Obj}}{\underline{D}}$ (here we consider ${\underline{E}}$ as a 