ScalingStacks

Proof. For $d\geq 1$ this is the content of T.2.3.4.12. For $d=-1$ this is trivial. For $d=0$, (1) and (2) are obvious from the definition. For (3) observe that we have a factorization of the map in question:

$$\operatorname{Fun}\left(h_{0}\mathcal{C},\mathcal{D}\right)\to\operatorname{Fun}\left(h_{1}\mathcal{C},\mathcal{D}\right)\overset{\sim}{\longrightarrow}\operatorname{Fun}\left(\mathcal{C},\mathcal{D}\right),$$

where the second map is an isomorphism (from the claim for $d=1$). Therefore, we can assume that $\mathcal{C}$ is an ordinary category and $\mathcal{D}$ is a poset and hence both simplicial sets are discrete. The result now follows from the observation that every functor $\mathcal{C}\to\mathcal{D}$ factors uniquely through $h_{0}\mathcal{C}$. ∎