  In this work we introduce the concept of generalized exponential
$\mathfrak{D}$-pullback attractor for evolution processes, where $\mathfrak{D}$
is a universe of families in $X$, which is a compact and positively invariant
family that pullback attracts all elements of $\mathfrak{D}$ with an
exponential rate. Such concept was introduced in arXiv:2311.15630 for the
general case of decaying functions (which include the exponential decay), but
for fixed bounded sets rather than to universe of families. We prove a result
that ensures the existence of a generalized exponential
$\mathfrak{D}_{\mathcal{C}^\ast}$-pullback attractor for an evolution process,
where $\mathfrak{D}_{\mathcal{C}^\ast}$ is a specific universe. This required
an adaptation of the results of arXiv:2311.15630, which only covered the case
of a polynomial rate of attraction, for fixed bounded sets. Later, we prove
that a nonautonomous wave equation has a generalized exponential
$\mathfrak{D}_{\mathcal{C}^\ast}$-pullback attractor. This, in turn, also
implies the existence of the $\mathfrak{D}_{\mathcal{C}^\ast}$-pullback
attractor for such problem.
