Kerodon

Remark 10.3.3.2. In the situation of Definition 10.3.3.1, our assumption that $i$ is a monomorphism guarantees that the composition map $\operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}( X, Y_0 ) \xrightarrow { [i] \circ } \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}( X, Y )$ is injective. It follows that if there exists a morphism $q: X \rightarrow Y_0$ satisfying $[f] = [i] \circ [q]$, then $q$ is uniquely determined up to homotopy. In particular, the condition that $q$ is a quotient morphism is independent of the choice of $q$ (see Example 10.3.2.13).