  In an abelian category $\mathscr{A}$ with small ${\rm Ext}$ groups, we show
that there exists a one-to-one correspondence between any two of the following:
balanced pairs, subfunctors $\mathcal{F}$ of ${\rm Ext}^{1}_{\mathscr{A}}(-,-)$
such that $\mathscr{A}$ has enough $\mathcal{F}$-projectives and enough
$\mathcal{F}$-injectives and Quillen exact structures $\mathcal{E}$ with enough
$\mathcal{E}$-projectives and enough $\mathcal{E}$-injectives. In this case, we
get a strengthened version of the translation of the Wakamatsu lemma to the
exact context, and also prove that subcategories which are
$\mathcal{E}$-resolving and epimorphic precovering with kernels in their right
$\mathcal{E}$-orthogonal class and subcategories which are
$\mathcal{E}$-coresolving and monomorphic preenveloping with cokernels in their
left $\mathcal{E}$-orthogonal class are determined by each other. Then we apply
these results to construct some (pre)enveloping and (pre)covering classes and
complete hereditary $\mathcal{E}$-cotorsion pairs in the module category.
