The passage from any small category **C** to its set-valued functor categoy
$\hat{\mathbf C}:=\mathrm{Fun}(C^{\ast},\mathrm{\bf Set})$ i.e.
the full Yoneda-embedding $Y\colon \mathrm{\bf C} \to \hat{\mathrm{\bf C}}$ into
the presheaf category can be considered as an universal completion-process.

A functor category such as $\mathrm{Fun}(\mathrm{\bf C}^{\ast},\mathrm{\bf Set})$ is a category which is ``almost as good as the target category $\mathrm{\bf Set}$''. In particular such a functor category is a topos and has an injective subobject classifier $\Omega$.

In the simplest case of $\mathrm{\bf Set}$, which can be considered as
the special case $\mathrm{\bf C}=\mathrm{\bf 1}$, the subobject classifier
is a two-element set $\{0,1\}$ and has the property, that it is a
cogenerator. (An object $C$ of a category is called *cogenerator*, if
for any two distinct morphism $f,g\colon X\to Y$ there is a morphism
$s\colon Y\to C$ such that $s\circ f\neq s\circ g$). A cogenerator $C$ allows
to separate the morphisms in the category and so to ``resolve''

the category, if $C$ happens to be an injective object.

It seem to be natural to ask whether the subobject classifier of any set-valued functor category is a cogenerator. This can not be the case, since in the special case $\mathrm{\bf C}={\mathbb Z}_2$ (category with one object generated by an non-trivial involution) which leads to the functor category $\hat{\mathbb Z}_2=\mbox{Fun}({\mathbb Z}_2^{\ast},\mathrm{\bf Set})$ of sets with a ${\mathbb Z}_2$-action, the subobject classifier $\Omega$ is a two-element set $\{0,1\}$ with the trivial ${\mathbb Z}_2$-action. This object $\Omega$ is not a cogenerator of $\hat{\mathbb Z}_2$ since the the two-element set with the nontrivial ${\mathbb Z}_2$-action gives an object $X$ of $\hat{\mathbb Z}_2$, whose nontrivial automorphism cannot be separated from the identity by any morphism from $X$ to $\Omega$. Hence there are conditions needed before the subobject classifier of a set-valued functor category can be a cogenerator.

Under which precise conditions for the category $\mathrm{\bf C}$ is the subobject classifier $\Omega$ of its free-cocompletion $\hat{\mathrm{\bf C}}$ a cogenerator?

"When is $\Omega$ a cogenerator in a topos?", available here: numdam.org/item?id=CTGDC_1975__16_1_3_0 $\endgroup$ – Theo Buehler May 10 '11 at 11:45