Remark 1.4.6.8. Let $f: X \rightarrow Y$ be a morphism in the $\infty $-category $\operatorname{\mathcal{C}}$. It follows from Remark 1.4.6.7 that the following conditions are equivalent:
- $(1)$
The morphism $f$ is an isomorphism.
- $(2)$
The morphism $f$ admits a homotopy inverse $g$.
- $(3)$
The morphism $f$ admits both left and right homotopy inverses.
In this case, the morphism $g$ is uniquely determined up to homotopy; moreover, any left or right homotopy inverse of $f$ is homotopic to $g$. We will sometimes abuse notation by writing $f^{-1}$ to denote a homotopy inverse to $f$.