Kerodon

Remark 6.2.2.3. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ be a full subcategory, and suppose we are given a pair of morphisms $u: X \rightarrow Y$ and $w: X \rightarrow Z$ of $\operatorname{\mathcal{C}}$, where $Y$ and $Z$ belong to the subcategory $\operatorname{\mathcal{C}}'$. If $u$ exhibits $Y$ as a $\operatorname{\mathcal{C}}'$-reflection of $X$, then we can realize $w$ as a composition of $u$ with another morphism $v: Y \rightarrow Z$ of $\operatorname{\mathcal{C}}'$, which is uniquely determined up to homotopy. Moreover, $v$ is an isomorphism if and only if $w$ exhibits $Z$ as a $\operatorname{\mathcal{C}}'$-reflection of $X$. Stated more informally: a $\operatorname{\mathcal{C}}'$-reflection of $X$, if it exists, is unique up to isomorphism.