Definition 1.4.4.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Suppose we are given objects $X,Y,Z \in \operatorname{\mathcal{C}}$ and morphisms $f: X \rightarrow Y$, $g: Y \rightarrow Z$, and $h: X \rightarrow Z$. We will say that *$h$ is a composition of $f$ and $g$* if there exists a $2$-simplex $\sigma $ of $\operatorname{\mathcal{C}}$ satisfying $d^{2}_0(\sigma ) = g$, $d^{2}_1(\sigma ) = h$, and $d^{2}_2(\sigma ) = f$. In this case, we will also say that the $2$-simplex $\sigma $ *witnesses $h$ as a composition of $f$ and $g$*.

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