Subobject Classifier of a Topos is Injective https://ift.tt/eA8V8J

Let $\mathcal{E}$ be a topos with subobject classifier $1\overset{t}{\rightarrow}\Omega$, then I want to show that $\Omega$ is injective.

So let $f:A\rightarrow\Omega$ and $g:A\rightarrow B$ be two maps in $\mathcal{E}$ with $g$ monic. Then I want to show that there exists a map $h:B\rightarrow\Omega$ such that $h\circ g = f$. Notice that by the property of a subobject classifier we have that there exists a unique map $\phi:B\rightarrow\Omega$ such that there is a pullback square $\require{AMScd}$ \begin{CD} A @>{\psi}>> 1\\ @VV{g}V @VV{t}V\\ B @>{\phi}>> \Omega \end{CD} Now I claim that $\phi$ does the job for $h$. But I don't see why $\phi\circ g= f$ (so maybe my claim is wrong). Any help would be appreciated!

from Hot Weekly Questions - Mathematics Stack Exchange
Peter