Every Map $X \xrightarrow{f} \Omega$ is the Characteristic Map of a Monomorphism
Proposition Let $\mathcal{E}$ be a topos with subobject classifier $\Omega$. If $f: X \to \Omega$, then $f = \chi_m$ for some mono $m: Y \rightarrowtail X$. Proof A topos is closed under taking pullbacks. We already have the diagram, Taking its pullback, we get a diagram, But a pullback of $\text{true}$ along any morphism $f$ is always monic, By the definition of subobject classifier, namely the part about uniqueness of $\chi_m$, we must have that $\chi_m = f$. $\blacksquare$