In a Topos, a Subobject Classifier is an Injective Object

Proposition 

Let $\mathcal{E}$ be a topos.  Then its subobject classifier $\Omega$ is an injective object.

Proof:

Let $f: X \to \Omega$ and $m: X \rightarrowtail Y$ be two maps in $\mathcal{E}$.  Then since every map $f:X \to \Omega$ is the characteristic map of some mono in $\mathcal{E}$ we have the following diagram,

where $S \in \mathcal{E}$ is some object, and $m_f$ is the monomorphism induced by $f$. Now, composing $m$ and $m_f$ we get another monomorphism $m'$. Now, draw the following diagram:

We know that a square of three monos and $\text{id}$ is a pullback square, therefore the inner square on the left is a pullback, in the above diagram. The right inner square is also a pullback, because $\Omega$ is a subobject classifier and $\chi_{m'}$ is the characteristic function of the mono $m'$.

By the pullback-pasting lemma, we have that the perimeter of the rectangle is a pullback square. This means:

is a pullback square. By the uniqueness part of the definition of subobject classifier, we have that $\chi_{m'} m = f$ since $f$ is aslo a characteristic map. That means there exists a morphism, namely $g = \chi_{m'} : Y \to \Omega$ such that $gm = f$, or that $\Omega$ is an injective object.

$\blacksquare$

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