Power Objects $PX$ are Injective

Proposition

Let $\mathcal{E}$ be a topos and $X \in \mathcal{E}$ an object. If $PX$ is a power object of $X$, then $PX$ is injective.

Proof

By definition of power object, we have $\text{Hom}_{\mathcal{E}}(X \times Y, \Omega) \simeq \text{Hom}_{\mathcal{E}}(X\times Z, \Omega)$ where the isomorphism is in $\textbf{Set}$ (homsets are <i>sets</i>!), for any $X,Y,Z \in \mathcal{E}$.  

Take 

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