In the category of sets, each *nonempty* set is injective since given a mono $A\ \rightarrowtail B$ and an arrow $A\rightarrow C$ we can lift to an arrow $B\rightarrow C$ by giving up injectivity : send each $a\in A\subset B$ to its image in $C$ and do whatever you want with the other elements of $B$.

So for sets injectivity is very simple. Now if one can think of an elementary topos as a category of "variable sets" (whatever that means), is there some sort of "dynamic" analogue for the intuition in the case of the category of sets? Maybe an analogous method of proof?

What's a ("the"?) simple intuitive proof that an elementary topos has enough injectives?