ScalingStacks

Proof. Since $p$ is a pullback of the right fibration $\left(\mathcal{O}_{\operatorname{\scriptsize{act}}}^{\otimes}\right)_{/X}\to\mathcal{O}_{\operatorname{\scriptsize{act}}}^{\otimes}$ it is itself a right fibration. The simplicial set $\mathbf{Triv}_{\operatorname{\scriptsize{act}}}^{\otimes}$ is isomorphic to $\mathbf{Fin}^{\simeq}$ and is in particular a Kan complex. By T.2.1.3.3 the map $p$ is a Kan fibration. ∎