This is Lemma 10.131.6 in the stack project.
It is mentioned here that we can select some $a^{'} \in S^{'}$ s.t. $a^{'} = \varphi(a_{i})$ for all i.
I would like to ask how this $a^{'}$ was found or, in other words, how to prove the existence of $a^{'}$. Meanwhile, does the proposition refer to whether $ker$ can be generated by one element $da$ or by all elements that meet the conditions.