Kerodon

Remark 7.7.1.18. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $Y \in \operatorname{\mathcal{C}}$ be a universal initial object (in the sense of Example 7.7.1.17). Then any morphism $f: Y \rightarrow Z$ in $\operatorname{\mathcal{C}}$ is a monomorphism. That is, if $X$ is another object of $\operatorname{\mathcal{C}}$, then the composition map $\operatorname{Hom}_{ \operatorname{\mathcal{C}}}( X, Y ) \xrightarrow { [f] \circ } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z )$ induces a homotopy equivalence from $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ to a summand of $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X, Z)$. This follows the observation that if the morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ is nonempty, then $X$ is isomorphic to $Y$; it follows that $X$ is also initial, so that the morphism spaces $\operatorname{Hom}_{ \operatorname{\mathcal{C}}}( X, Y )$ and $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z )$ are both contractible.