§\[sws\] of the paper where a rich collection of properties of pure functors and [*virtual $t$-truncations*]{} of (co)homological functors (with respect to $w$) is proved.

Now, virtual $t$-truncations are defined in terms of weight structures; still they are closely related to $t$-structures (whence the name). Respectively, our results yield the existence of some vast new families of $t$-structures. To describe one of the main results of this sort here we recall that a $t$-structure $t=({\underline{C}}^{t\le 0}, {\underline{C}}^{t\ge 0})$ (for a triangulated category ${\underline{C}}$) is said to be (right) adjacent to $w$ if ${\underline{C}}^{t\le 0}={\underline{C}}_{w\ge 0}$.[^3] For a triangulated category ${\underline{C}}$ that is closed with respect to (small) coproducts and a weight structure $w$ on it we will say that $w$ is [*smashing*]{} whenever ${\underline{C}}_{w\ge 0}$ is closed with respect to ${\underline{C}}$-coproducts (note that ${\underline{C}}_{w\le 0}$ is $\coprod$-closed automatically).

\[tadjti\] Let ${\underline{C}}$ be a triangulated category that is closed with respect to coproducts and satisfies the following Brown representability property: any functor ${\underline{C}}{{^{op}}}\to {\underline{\operatorname{Ab}}}$ that respects (${\underline{C}}{{^{op}}}$)-coproducts is representable.[^4]

Then for a weight structure $w$ on ${\underline{C}}$ there exists a $t$-structure right adjacent to it if and only if $w$ is smashing. Moreover, the heart of $t$ (if $t$ exists) is 