Equalizer

Given in a Category an equalizer of is an Object and map such that and for any if we have a unique making the Commutative Diagram

\usepackage{tikz-cd}
\begin{document}
\begin{tikzcd}
G \arrow[dr, "j"] \arrow[d, "s"]\\
E \arrow[r, "i"] 
& A \arrow[r, "f{,}g"] 
& B
\end{tikzcd}
\end{document}

Note that is a Monomorphism and we say that in this case is a Regular Monomorphism

Lemma

If has equalizers, then any Detecting Family in is also a Separating Family.

Proof

Suppose is a Detecting Family. Let be such that for any with Let be an equalizer of and . Then there is a unique such that , so factors uniquely through and hence is an Isomorphism. But so and hence is a Separating Family.