  We compute the (stable) \'etale cohomology of $\mathrm{Hom}_{n}(C,
\mathcal{P}(\vec{\lambda}))$, the moduli stack of degree $n$ morphisms from a
smooth projective curve $C$ to the weighted projective stack
$\mathcal{P}(\vec{\lambda})$, the latter being a stacky quotient defined by
$\mathcal{P}(\vec{\lambda}) := \left[\mathbb{A}^N-\{0\}/\mathbb{G}_m\right]$,
where $\mathbb{G}_m$ acts by weights $\vec{\lambda} = (\lambda_0, \cdots,
\lambda_N) \in \mathbb{Z}^N_{+}$. Our key ingredient is formulating and proving
the \'etale cohomological descent over the category $\Delta S$, the symmetric
(semi)simplicial category. An immediate arithmetic consequence is the
resolution of the geometric Batyrev--Manin type conjecture for weighted
projective stacks over global function fields. Along the way, we also analyze
the intersection theory on weighted projectivizations of vector bundles on
smooth Deligne-Mumford stacks.
