Definition 2.7 (Separating / detecting family). Let $G$ be a family of objects of a locally small category $C$.

• (a)
  We say $G$ is a separating family if the functors $C(G,\bullet )$, $G\in G$ are jointly faithful, i.e. given a parallel pair $A\underset{g}{\overset{f}{\rightrightarrows }}B$, the equations $fh=gh$ for all $h:G\to A$ with $G\in G$ imply $f=g$.
• (b)
  We say $G$ is a detecting family if the $G(G,\bullet )$ jointly reflect isomorphisms, i.e. given $A\stackrel{f}{\to }B$, if every $G\stackrel{g}{\to }B$ with $G\in G$ factors uniquely through $f$, then $f$ is an isomorphism.

If $G=\{G\}$, we call $G$ a separator or a detector.