equalizer in Set
1. Proposition
Given the category set, sets \(A,B \in \mathrm{Ob}(\mathrm{Set})\) and maps \(\varphi, \psi: A \rightarrow B\), the equalizer is the set
\begin{align*} E \coloneqq \{a \in A \vert \varphi(a) = \psi(a)\} \subseteq A \end{align*}with the usual inclusion