Maps to a Subobject Classifier Let $\Omega$ be the subobject classifier of a category $C$. Choose some map $m: X \rightarrow \Omega$. Is there necessarily a subobject $p: P \rightarrow X$ for which $m$ is the characteristic map?  
It seems that in the definition of subobject classifier you can only go the other way around, i.e. from subobjects of $X$ to characteristic maps so that there may be more maps than subobjects. Is that possible?
 A: The definition of subobject classifier is that it represents the $\mathsf{Sub}$ functor.  That is, there is a natural bijection $\text{Hom}(-,\Omega) \cong \mathsf{Sub}$.  So you can go either way. Given a subobject of $X$, i.e. an element of $\mathsf{Sub}(X)$, you have an arrow $X\to\Omega$ and vice versa. Given $\chi : X \to \Omega$, you find $m : P \rightarrowtail X$ by pulling back $\chi$ along $\mathsf{true} : 1 \rightarrowtail\Omega$.
I should note that for the $\mathsf{Sub}$ functor to even exist as a functor $\mathcal{C}^{op}\to\mathbf{Set}$, it's necessary for $\mathcal{C}$ to be well-powered (otherwise $\mathsf{Sub}$ wouldn't be $\mathbf{Set}$-valued) and for pullbacks along monomorphisms to exist (otherwise it wouldn't have an action on arrows).
A: Usually you only speak of a subobject classifier in a category that has all pullbacks.  In that case, the pullback of the universal map $1\to\Omega$ along $m$ is a subobject $p:P\to X$ for which $m$ is the characteristic map.  Really, the existence of at least all pullbacks of this form should be part of the definition of a subobject classifier, but it is sometimes not mentioned since the definition is only used in categories which have all pullbacks.
If you don't assume any pullbacks exist, then such a subobject need not exist.  For instance, consider the category of nonempty sets.  The usual subobject classifier for sets still satisfies the definition "every subobject has a unique characteristic map" and so gives a "subobject classifier" in this category by that definition.  But a function which is the characteristic function of the empty subset is not the characteristic function of any subobject in this category.
