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 $\chi_m = f$.

$\blacksquare$

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