be described “more explicitly” in terms of ${\mathcal{P}}$; cf. Corollary \[cwftw\] below.

Thus if we assume in addition that the Brown representability condition is fulfilled for ${\underline{C}}$[^12]then we also obtain the (right) adjacent $t$-structure $t=(R, (R{{}^{\perp}})[1])$. This $t$-structure is [*cogenerated*]{} by any class ${\mathcal{P}}'$ that is [*weakly symmetric*]{} to ${\mathcal{P}}$ (see Definition \[dsym\](\[iwsym\]) below; if the elements of ${\mathcal{P}}$ are compact then we can construct ${\mathcal{P}}'$ using [*Brown-Comenetz duality*]{}). Somewhat surprisingly to the author, a similar chain of arguments gives the existence of a weight structure $w$ that is [*right adjacent*]{} to a given compactly generated $t$-structure (i.e., ${\underline{C}}^{t\ge 0}={\underline{C}}_{w\le 0}$); this yields the proof of Theorem \[tgroth\](1).[^13] Moreover, the “opposite” weight structure $w{{^{op}}}$ in the category ${\underline{C}}{{^{op}}}$ is perfectly generated but not compactly generated (so, there exist plenty of perfectly generated weight structures that are not compactly generated; recall that the latter class of weight structures was introduced in [@paucomp]). We also prove the following “well generatedness” result for weight structures (obtaining in particular that all smashing weight structures on well generated categories can be obtained from Theorem \[textw\]).

\[pwgwstr\] Assume that ${\underline{C}}$ is a well generated triangulated category (i.e., there exists a [*regular*]{} cardinal ${\alpha}$ and a perfect 