Proposition 1.6.8 of [@neebook], and the second one is is given by Proposition 1.2.1 and Remark 1.2.2 of ibid.

\[dcomp\] Assume that ${\underline{C}}$ has coproducts; ${\mathcal{P}}\subset {\operatorname{Obj}}{\underline{C}}$.

1.  \[idcompa\] An object $M$ of ${\underline{C}}$ is said to be [*compact*]{} if the functor $H^M={\underline{C}}(M,-):{\underline{C}}\to {\underline{\operatorname{Ab}}}$ respects coproducts.

2.  \[idloc\] For ${\underline{D}}\subset {\underline{C}}$ (${\underline{D}}$ is a triangulated category that may be equal to ${\underline{C}}$) one says that ${\mathcal{P}}$ generates ${\underline{D}}$ [*as a localizing subcategory*]{} of ${\underline{C}}$ if ${\underline{D}}$ is the smallest full strict triangulated subcategory of ${\underline{C}}$ that contains ${\mathcal{P}}$ and is closed with respect to ${\underline{C}}$-coproducts.

    If this is the case then we will also say that ${\mathcal{P}}$ [*cogenerates*]{} ${\underline{D}}{{^{op}}}$ as a [*colocalizing*]{} subcategory of ${\underline{C}}{{^{op}}}$.

3.  \[idhg\] We will say that ${\mathcal{P}}$ [*Hom-generates*]{} a full triangulated category ${\underline{D}}$ of ${\underline{C}}$ containing ${\mathcal{P}}$ if ${\operatorname{Obj}}{\underline{D}}\cap(\cup_{i\in {{\mathbb{Z}}}}{\mathcal{P}}[i]){{}^{\perp}}={\{0\}}$.

4.  \[idcg\] We will say that ${\underline{C}}$ is [*compactly generated*]{} if it is Hom-generated by a set of compact objects.

5.  \[idcc\] It will be convenient for us to use the following somewhat clumsy terminology: a homological functor $H:{\underline{C}}\to {\underline{A}}$ (where ${\underline{A}}$ is an abelian category) will be called a [*cc*]{} functor if it respects coproducts (i.e., the image of any coproduct in ${\underline{C}}$ is 