ScalingStacks

Proposition 3.1.4 ([SY19, Theorem 2.15]). The inclusion $\mathbf{Cat}_{d}\hookrightarrow\mathbf{Cat}_{\infty}$ admits a left adjoint $h_{d}$, such that for every $\infty$-category $\mathcal{C}$, the value of $h_{d}$ on $\mathcal{C}$ is the $d$-homotopy category of $\mathcal{C}$, the unit transformation $\theta_{d}\colon\mathcal{C}\to h_{d}\mathcal{C}$ is essentially surjective, and for all $X,Y\in\mathcal{C}$, the map of spaces

$$\operatorname{Map}_{\mathcal{C}}\left(X,Y\right)\to\operatorname{Map}_{h_{d}\mathcal{C}}\left(\theta_{d}\left(X\right),\theta_{d}\left(Y\right)\right)$$

is the $\left(d-1\right)$-truncation map.