In a topos with a NNO, how does one define the subobject "$<$'' of $\mathbf N\times\mathbf N$?

The definition of $<$ in terms of addition, as in the last paragraph of Derek Elkins's answer, seems to be a favorite of many topos theorists. My own preference is to define $x<y$ by induction on $y$. (Formally, that means defining the curried version $\mathbb N\to\Omega^{\mathbb N}:y\mapsto(x\mapsto [x<y]$ of the predicate $<:\mathbb N^2\to\Omega$.) The inductive clauses are $$ [x<0] = \bot $$ and $$ [x<s(y)]=([x<y]\lor[x=y]). $$