Remark 1.5.7.8 (The Generalized Associative Law). Let $\operatorname{\mathcal{C}}$ be an ordinary category and let $n \geq 0$ be an integer. Applying Remark 1.5.7.4 to the inner anodyne inclusion $\operatorname{Spine}[n] \hookrightarrow \Delta ^ n$ of Example 1.5.7.7, we deduce that every diagram
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 1.5.7.3 can be regarded as an extension of this generalized associative law to the setting of $\infty $-categories.