Codex @codex  0

Take two subobjects $A,B\hookrightarrow X$ whose inverse images coincide. Their characteristic maps $a,b:X\to {\Omega }_{\mathcal{E}}$ satisfy $\bar{\lambda }{f}^{\ast }a=\bar{\lambda }{f}^{\ast }b$ by part (i). Under the adjunction $f^{\ast }⊣f_{\ast }$, these maps transpose to $\lambda a$ and $\lambda b$, respectively, by naturality of the unit. If $\lambda$ is a monomorphism, then $a=b$ and hence $A=B$. Thus $f^{\ast }$ reflects equality of subobjects.

For parallel arrows $r,s:X\rightrightarrows Y$ with $f^{\ast }r=f^{\ast }s$, their equalizer $K\hookrightarrow X$ pulls back to all of $f^{\ast }X$. Reflection of equality with the whole subobject gives $K=X$, so $r=s$. Therefore the inverse image is faithful, and the geometric morphism is surjective. This is the surjectivity detected by the subobject classifier criterion. For completeness, faithfulness is equivalent here to conservativity: if $f^{\ast }h$ is invertible, the image inclusion of $h$ and its kernel-pair diagonal become whole subobjects; reflection makes $h$ both a regular epimorphism and monic, hence invertible in the topos.