ScalingStacks

Lemma 2.8 (T.2.3.4.12). For $d\geq 1$, given an $\infty$-category $\mathcal{C}$, there exists an essentially unique simplicial set $h_{d}\mathcal{C}$, such that for every simplicial set $K$, we have a bijection

$$\hom\left(K,h_{d}\mathcal{C}\right)\simeq\left[K^{d-1},K^{d},K^{d+1};\mathcal{C}\right]$$

that is natural in $K$. We denote the canonical map by $\theta_{d}\colon\mathcal{C}\to h_{d}\mathcal{C}$.