equivalent to the category of all those additive functors ${{\underline{Hw}}}{{^{op}}}\to {\underline{\operatorname{Ab}}}$ that respect products.[^5]

Note here that triangulated categories closed with respect to coproducts have recently become very popular in homological algebra and found applications in various areas of mathematics; so a significant part of this paper is dedicated to categories of this sort. Still we prove certain alternative versions of Theorem \[tadjti\]; some of them can be applied to “quite small” triangulated categories.

So, if instead of the Brown representability condition for ${\underline{C}}$ we demand it to satisfy the [*$R$-saturatedness one*]{} (see Definition \[dsatur\] below; this is an “$R$-linear finite” version of the Brown representability) then for any [*bounded*]{} $w$ on ${\underline{C}}$ there will exist a $t$-structure right adjacent to it. Note that this version of the result can be applied to the bounded derived category $D^b(X)$ of coherent sheaves on regular separated finite-dimensional scheme that is proper over the spectrum of a Noetherian ring $R$ (see Proposition \[psatur\](II) and Remark \[rsatur\](1)). We also prove two generalizations of this existence result (see Proposition \[psaturdu\] and Remark \[roq\]); they “produce” certain $t$-structures from weight structures on the bounded derived category of coherent sheaves on any proper $R$-scheme $X$ and also 