Remark 3.1.6.4. Let $f: X_{} \rightarrow Y_{}$ be a morphism of simplicial sets. The condition that $f$ is a homotopy equivalence depends only on the homotopy class $[f] \in \pi _0( \operatorname{Fun}(X_{}, Y_{} ) )$. Moreover, if $f$ is a homotopy equivalence, then its simplicial homotopy inverse $g: Y_{} \rightarrow X_{}$ is determined uniquely up to homotopy.
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