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Definition Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f,g: C \rightarrow D$ be a pair of morphisms in $\operatorname{\mathcal{C}}$ having the same source and target. A homotopy from $f$ to $g$ is a $2$-simplex $\sigma $ of $\operatorname{\mathcal{C}}$ satisfying $d_0( \sigma ) = \operatorname{id}_{D}$, $d_1(\sigma ) = g$, and $d_2(\sigma ) = f$, as depicted in the diagram

\[ \xymatrix { & D \ar [dr]^{\operatorname{id}_ D} & \\ C \ar [ur]^{f} \ar [rr]^{g} & & D. } \]

We will say that $f$ and $g$ are homotopic if there exists a homotopy from $f$ to $g$.