Kerodon

Remark 3.2.4.19. Let $X$ be a simplicial set. Then $X$ is weakly contractible if and only if, for every Kan complex $Y$, every morphism of simplicial sets $f: X \rightarrow Y$ is nullhomotopic. To prove this, we may assume that $X$ is nonempty (otherwise the identity morphism $\operatorname{id}_{X}$ is not nullhomotopic; see Example 3.2.4.6). Then, for any Kan complex $Y$, the diagonal map $\delta _{Y}: Y \rightarrow \operatorname{Fun}(X,Y)$ admits a left inverse (given by evaluation at any vertex $x \in X$), and is automatically injective on connected components. It follows that $X$ is weakly contractible if and only if, for every Kan complex $Y$, the morphism $\delta _{Y}$ is also surjective on connected components: that is, every morphism $f: X \rightarrow Y$ is homotopic to a constant map.