  Let $R\to A$ be a homomorphism of associative rings, and let $(\mathcal
F,\mathcal C)$ be a hereditary complete cotorsion pair in $R\mathsf{-Mod}$. Let
$(\mathcal F_A,\mathcal C_A)$ be the cotorsion pair in $A\mathsf{-Mod}$ in
which $\mathcal F_A$ is the class of all left $A$-modules whose underlying
$R$-modules belong to $\mathcal F$. Assuming that the $\mathcal F$-resolution
dimension of every left $R$-module is finite and the class $\mathcal F$ is
preserved by the coinduction functor $\operatorname{Hom}_R(A,-)$, we show that
$\mathcal C_A$ is the class of all direct summands of left $A$-modules finitely
(co)filtered by $A$-modules coinduced from $R$-modules from $\mathcal C$.
Assuming that the class $\mathcal F$ is closed under countable products and
preserved by the functor $\operatorname{Hom}_R(A,-)$, we prove that $\mathcal
C_A$ is the class of all direct summands of left $A$-modules cofiltered by
$A$-modules coinduced from $R$-modules from $\mathcal C$, with the decreasing
filtration indexed by the natural numbers. A combined result, based on the
assumption that countable products of modules from $\mathcal F$ have finite
$\mathcal F$-resolution dimension bounded by $k$, involves cofiltrations
indexed by the ordinal $\omega+k$. The dual results also hold, provable by the
same technique going back to the author's monograph on semi-infinite
homological algebra arXiv:0708.3398. In addition, we discuss the $n$-cotilting
and $n$-tilting cotorsion pairs, for which we obtain better results using a
suitable version of a classical Bongartz-Ringel lemma. As an illustration of
the main results of the paper, we consider certain cotorsion pairs related to
the contraderived and coderived categories of curved DG-modules.
