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Remark (The Generalized Associative Law). Let $\operatorname{\mathcal{C}}$ be an ordinary category and let $n \geq 0$ be an integer. Applying Remark to the inner anodyne inclusion $\operatorname{Spine}[n] \hookrightarrow \Delta ^ n$ of Example, we deduce that every diagram

\[ X_0 \xrightarrow {f_1} X_1 \xrightarrow { f_2} X_2 \rightarrow \cdots \xrightarrow {f_ n} X_ n \]

can be extended uniquely to a functor $[n] \rightarrow \operatorname{\mathcal{C}}$. In particular, it shows that $\operatorname{\mathcal{C}}$ satisfies the “generalized associative law”: the iterated composition $f_{n} \circ f_{n-1} \circ \cdots \circ f_{2} \circ f_{1}$ is well-defined (that is, it does not depend on a choice of parenthesization). In essence, Proposition can be regarded as an extension of this generalized associative law to the setting of $\infty $-categories.