Remark 1.4.6.5. Let $f: X \rightarrow Y$ and $g: Y \rightarrow X$ be morphisms in an $\infty $-category $\operatorname{\mathcal{C}}$. Then the condition that $g$ is a left homotopy inverse (right homotopy inverse, homotopy inverse) to $f$ depends only on the homotopy classes $[f]$ and $[g]$.
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