To simplify our notation, we take the following conventions when studying a class \(S\) of morphisms in an \(\infty\)-category \(\mathcal C\).
Assuming that \(S\) consists of precisely the morphisms in a subcategory of \(\mathcal C\) (e.g. both classes in a factorization system on \(\mathcal C\)), we simply write \(S\) to denote this subcategory.
Assuming that \(S\) is stable under homotopy (e.g. both classes in a factorization system on \(\mathcal C\)), we also simply write \(S\) to denote the full subcategory of \(\mathrm{Fun}([1],\mathcal C)\) on the morphisms in \(S\).
We simply write \(\mathcal C^\simeq\) for the class of equivalences in \(\mathcal C\), and we simply write \(\mathcal C\) for the class of all morphisms in \(\mathcal C\).
For any object \(c \in \mathcal C\), we write \({\mathcal C}_{\small{/^{S}}{c}} \subseteq \mathcal C_{/c}\) for the full subcategory on those objects \((d \rightarrow c) \in \mathcal C_{/c}\) that lie in \(S\) (when considered as morphisms in \(\mathcal C\)). In the special case that \(c \simeq {\sf pt}_\mathcal C\) is terminal, we simply write \(\mathcal C^S \coloneqq {\mathcal C}_{\small{/^{S}}{{\sf pt}_{\mathcal C}}}\).