ScalingStacks

Proof. The functor $\sigma_{!}$ is a left adjoint, hence it preserves all colimits in $\pre(\Theta_{n})$, in particular pushouts. Moreover, $\sigma_{!}$ sends the generators of $T_{\Theta_{n-1}}$ to generators of $T_{\Theta_{n}}$. Together these imply the containment $\sigma_{!}(T_{\Theta_{n-1}})\subseteq T_{\Theta_{n}}$. Direct computations, which we leave to the reader, show that $\sigma_{!}$ sends $T_{\Theta_{n-1}}$-local objects to $T_{\Theta_{n}}$-local objects and that the following formula holds,

$$\hom(([n];o_{1},\dots,o_{n}),\sigma(X\times_{Y}Z))=\coprod_{i_{k}:[n]\to[1]\atop 0\leq k\leq n+1}\map(o_{k},X\times_{Y}Z).$$

where $i_{k}:[n]\to[1]$ maps $i\in[n]$ to $0\in[1]$ if $i<k$ and to $1\in[1]$ otherwise. In the above formula when $k=0$ or $n+1$ the space $\map(o_{k},X\times_{Y}Z)$ is interpreted as a singleton space. From this it follows that $\sigma$ preserves fiber products. ∎