  A classic result by Bass says that the class of all projective modules is
covering, if and only if it is closed under direct limits. Enochs extended the
if-part by showing that every class of modules $\mathcal C$, which is
precovering and closed under direct limits, is covering, and asked whether the
converse is true. We employ the tools developed in [18] and give a positive
answer when $\mathcal C = \mathcal A$, or $\mathcal C$ is the class of all
locally $\mathcal A ^{\leq \omega}$-free modules, where $\mathcal A$ is any
class of modules fitting in a cotorsion pair $(\mathcal A, \mathcal B)$ such
that $\mathcal B$ is closed under direct limits. This setting includes all
cotorsion pairs and classes of locally free modules arising in
(infinite-dimensional) tilting theory. We also consider two particular
applications: to pure-semisimple rings, and artin algebras of infinite
representation type.
