nLab
cogenerator

A cogenerator in a category $C$ is an object $S$ such that the functor $h_S=C(-,S):C^{\mathrm{op}}\mathrm{to}\mathrm{Set}$ is faithful. This means that for any pair $g_1,g_2\in C(X,Y)$, if they are indistinguishable by morphisms to $S$ in the sense that

$$\forall (\theta :Y\to S),\theta \circ g_1=\theta \circ g_2,$$\forall (\theta: Y \to S),\; \theta \circ g_1 = \theta \circ g_2 ,

then $g_1=g_2$.

One often extends this notion to a cogenerating family of objects, which is a (usually small) set $𝒮=\{S_a,a\in A\}$ of objects in $C$ such that the family $C(-,S_a)$ is jointly faithful. This means that for any pair $g_1,g_2\in C(X,Y)$, if they are indistinguishable by morphisms to $𝒮$ in the sense that

$$\forall (a:A),\forall (\theta :X\to S_a),\theta \circ g_1=\theta \circ g_2,$$\forall (a: A),\; \forall (\theta: X \to S_a),\; \theta \circ g_1 = \theta \circ g_2 ,

then $g_1=g_2$.

The dual notion is generator.

Examples

In Set, the set of truth values is a cogenerator. More generally, in any well-pointed topos, the subobject classifier is a cogenerator.

The existence of a small (co)generating family is one of the conditions in one version of the adjoint functor theorem.