Proposition 11.5.0.30. Let $Y$ be a simplicial set. The following conditions are equivalent:
Every morphism of simplicial sets $f: X \rightarrow Y$ is nullhomotopic.
Every morphism of simplicial sets $g: Y \rightarrow Z$ is nullhomotopic.
The identity morphism $\operatorname{id}_{Y}: Y \rightarrow Y$ is nullhomotopic.
The simplicial set $Y$ is contractible.
Proof. The implications $(1) \Rightarrow (3)$ and $(2) \Rightarrow (3)$ are immediate, and the reverse implications follow from Remark 3.2.4.10. To see that $(3) \Leftrightarrow (4)$, it suffices to observe that a morphism $y: \Delta ^0 \rightarrow Y$ is homotopy inverse to the projection map $Y \rightarrow \Delta ^0$ if and only if the identity morphism $\operatorname{id}_{Y}$ is homotopic to the constant morphism $Y \twoheadrightarrow \{ y\} \hookrightarrow Y$. $\square$