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We compute the (stable) \'etale cohomology of $\mathrm{Hom}_{n}(C,\mathcal{P}(\vec{\lambda}))$, the moduli stack of degree $n$ morphisms from asmooth 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 provingthe \'etale cohomological descent over the category $\Delta S$, the symmetric(semi)simplicial category. An immediate arithmetic consequence is theresolution of the geometric Batyrev--Manin type conjecture for weightedprojective stacks over global function fields. Along the way, we also analyzethe intersection theory on weighted projectivizations of vector bundles onsmooth Deligne-Mumford stacks.

\'Etale cohomological stability of the moduli space of stable elliptic surfaces