their restrictions. So for ${\underline{D}}'$ being a full triangulated subcategory of a triangulated category ${\underline{D}}$ and $M\in {\operatorname{Obj}}{\underline{D}}$ we will often write $H^M: {\underline{D}}'\to {\underline{\operatorname{Ab}}}$ for the restriction of the corepresentable (homological) functor ${\underline{D}}(M,-)$ to ${\underline{D}}'$ (yet $H^P$ in Proposition \[porthop\] will denote a certain [*coextension*]{}); $H_M: {\underline{D}}'{{^{op}}}\to {\underline{\operatorname{Ab}}}$ is the restriction of the functor ${\underline{D}}(-,M)$ to ${\underline{D}}'$. We will often be interested in the case ${\underline{D}}'={\underline{D}}$ in this notation; we assume that the domain of the functors $H^M$ and $H_M$ is the category ${\underline{C}}$ if not specified otherwise.

For $X,Y\in {\operatorname{Obj}}{\underline{C}}$ we will write $X\perp Y$ if ${\underline{C}}(X,Y)={\{0\}}$. For $D,E\subset {\operatorname{Obj}}{\underline{C}}$ we will write $D\perp E$ if $X\perp Y$ for all $X\in D,\ Y\in E$. For $D\subset{\operatorname{Obj}}{\underline{C}}$ we will write $D^\perp$ for the class $$\{Y\in {\operatorname{Obj}}{\underline{C}}:\ X\perp Y\ \forall X\in D\};$$ sometimes we will write $\perp_{{\underline{C}}}$ instead to indicate the category that we are considering. Dually, ${}^\perp{}D$ is the class $\{Y\in {\operatorname{Obj}}{\underline{C}}:\ Y\perp X\ \forall X\in D\}$.

In this paper all complexes will be cohomological, i.e., the degree of all differentials is $+1$; respectively, we use cohomological notation for their terms.

We will use the term [*exact functor*]{} for a functor of triangulated categories (i.e., for a functor that 