Remark 1.2.6.6. In the proof of Proposition 1.2.6.5, we have implicitly invoked the fact that every category $\operatorname{\mathcal{C}}$ satisfies the *generalized associative law*: every sequence of composable morphisms

has a well-defined composition $f_ n \circ f_{n-1} \circ \cdots \circ f_1$, which can be computed in terms of the binary composition law by inserting parentheses arbitrarily. One might object that this logic is circular: the generalized associative law is essentially equivalent to Proposition 1.2.6.5 (applied to the directed graph $G$ described in Example 1.2.6.2). In ยง1.4.7, we will establish an $\infty $-categorical generalization of Proposition 1.2.6.5 (Theorem 1.4.7.1), whose proof will avoid this sort of circular reasoning (see Remark 1.4.7.4).