Kerodon

Remark 5.4.6.3. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $W$ be a collection of edges of $\operatorname{\mathcal{C}}$. We can informally summarize Definition 5.4.6.1 as follows: a collection of morphisms $W$ of $\operatorname{\mathcal{C}}$ has the two-out-of-six property if, for every triple of composable morphisms $f: A \rightarrow B$, $g: B \rightarrow C$, and $h: C \rightarrow D$, if the compositions $g \circ f$ and $h \circ g$ belong to $W$, then the morphisms $f$, $g$, $h$, and $h \circ g \circ f$ belong to $W$. Beware that this summary is somewhat imprecise, since the compositions $g \circ f$, $h \circ g$, and $h \circ g \circ f$ are a priori only well-defined up to homotopy.