Generated congruences

Let \(\Sigma\) be a small set of morphisms in a logos \(T\). The congruence generated by \(\Sigma\), denoted \(\Sigma^c\), is defined as the intersection of all strongly saturated classes containing \(\Sigma\) that are closed under base change. It is a non-trivial fact that \(\Sigma^c\) is of small generation (and thus a congruence in the sense of Definition 4.14). This is proved by Lurie (2009, Proposition 6.2.1.2). We will give an independent proof below, see Remark 5.42.