the corresponding coproduct in ${\underline{A}}$); $H$ will be called a [*wcc*]{} functor if it respects countable coproducts.

    A cohomological functor $H$ from ${\underline{C}}$ into ${\underline{A}}$ will be called a [*cp*]{} functor if it converts all (small) coproducts into ${\underline{A}}$-products.

    Dually, for a triangulated category ${\underline{D}}$ that has products we will call a homological functor $H:{\underline{D}}\to {\underline{A}}$ a [*pp*]{} functor if its respects products.

6.  \[idbrown\] We will say that ${\underline{C}}$ satisfies the [*Brown representability*]{} property whenever any cp functor from ${\underline{C}}$ into ${\underline{\operatorname{Ab}}}$ is representable.

    Dually, we will say that a triangulated category ${\underline{D}}$ satisfies the dual Brown representability property if it has products and any pp functor from ${\underline{C}}$ into ${\underline{\operatorname{Ab}}}$ is corepresentable (i.e., if ${\underline{D}}{{^{op}}}$ satisfies the Brown representability property).

\[pcomp\] Assume that ${\underline{C}}$ has coproducts. I. Let ${\mathcal{P}}\subset {\operatorname{Obj}}{\underline{C}}$.

1\. If ${\mathcal{P}}$ generates ${\underline{C}}$ as its own localizing subcategory then it also Hom-generates ${\underline{C}}$. Conversely, if ${\mathcal{P}}$ Hom-generates ${\underline{C}}$ and the embedding into ${\underline{C}}$ of its localizing subcategory ${\underline{C}}'$ generated by ${\mathcal{P}}$ possesses a right adjoint then ${\underline{C}}'={\underline{C}}$.

2\. More generally, denote by $C$ the smallest coproductive extension-closed subclass of ${\operatorname{Obj}}{\underline{C}}$ containing ${\mathcal{P}}$. Then ${\mathcal{P}}{{}^{\perp}}\cap C={\{0\}}$. II.1. Assume that ${\underline{C}}$ is compactly generated. Then both ${\underline{C}}$ and 