ScalingStacks

Proof. For $d=-1,0$, both assertions are trivial to check and so we assume that $d\geq 1$. The argument that $h_{d}\left(p\right)$ is an inner fibration is similar to the argument that $h_{d}\left(f\right)$ is coCartesian and so we shall prove them together. Using T.2.4.1.4, we need to consider the lifting problem

for some $m\geq 2$ and either

1. (1)

   $0<i<m$ or

2. (2)

   $i=0$ and $\Delta^{\left\{0,1\right\}}\subseteq\Lambda_{0}^{m}$ is mapped in $h_{d}\mathcal{C}$ to $h_{d}\left(f\right)$.

For $m\geq d+3$, we have $\mbox{sk}^{j}\Lambda_{i}^{m}=\mbox{sk}^{j}\Delta^{m}$ for all $j\leq d+1$, and so the map

$$\hom\left(\Delta^{m},h_{d}\mathcal{C}\right)\to\hom\left(\Lambda_{i}^{m},h_{d}\mathcal{C}\right)$$

is a bijection and there is nothing to prove. For $m\leq d+2$, we have $\Lambda_{i}^{m}=\mbox{sk}^{d+1}\Lambda_{i}^{m}$, and so the map

$$\hom\left(\Lambda_{i}^{m},\mathcal{C}\right)\twoheadrightarrow\hom\left(\Lambda_{i}^{m},h_{d}\mathcal{C}\right)$$

is surjective, hence the map $\Lambda_{i}^{m}\to h_{d}\mathcal{C}$ factors through $\Lambda_{i}^{m}\to\mathcal{C}$. Now, the functor $\mathcal{C}\to h_{d}\mathcal{C}$ identifies only homotopic morphisms (for $d\geq 1$); hence in (2) the image of $\Delta^{\left\{0,1\right\}}$ in $\mathcal{C}$ is coCartesian. Thus, in both cases we can solve the corresponding lifting problem in $\mathcal{C}$, which induces a lift in the original square. ∎