collection of generators in an infinity category

Let \(\mathcal{C}\) be a locally small infinity category, \(S \subseteq \mathcal{C}_0\) a collection of objects.
Then \(S\) is said to be a generator of \(\mathcal{C}\), if for any \(f: c \rightarrow d\) TFAE:

1. \(f_*: \mathrm{map}_{\mathcal{C}}(s,c) \rightarrow \mathrm{map}_{\mathcal{C}}(s,d)\) is an equivalence for each \(s \in S\)
   
2. \(f: c \rightarrow d\) is an equivalence
   

under set theoretic assumptions, e.g. \(S\) is small for in the universe with respect to \(\mathrm{An}\), this amounts to:

\begin{align*} \prod_{s \in S} \mathrm{map}_{\mathcal{C}}(s,-) \end{align*}

is a conservative