ScalingStacks

Proof. Let $G\colon\Delta[1]\rightarrow W_{1}$ denote the path connecting $g$ and $g^{\prime}$. Then it suffices to note that a dotted arrow exists in

where $H$ is a lift of $(s_{0}d_{1}g^{\prime},g^{\prime},s_{0}d_{0}g^{\prime})=(id_{x^{\prime}},g^{\prime},id_{y^{\prime}})$ to $W_{3}$, since the right-hand vertical map is a fibration. ∎