mp.mathematical physics - Morphisms of supermanifolds

Unlike many schemes, but similar to ordinary manifolds, a map of super-manifolds $$(X, mathcal{O}_X) to (Y, mathcal{O}_Y)$$ determines and is completely determined by the map of superalgebras obtained by looking at global sections:
$$mathcal{O}_Y(Y) to mathcal{O}_X(X)$$
In the example at hand this is the graded ring map:
$$ x mapsto x' + a'b'$$
$$ a mapsto a'$$
$$ b mapsto b'$$

This map induces a map of rings after we mod out by nilpotents:
$$C^infty(Y) = mathcal{O}_Y/Nil to mathcal{O}_X / Nil = C^infty(X)$$
This map in turn induces a smooth map $X to Y$ (in fact it is equivalent to such a map). In this case, after modding out by nilpotents we get the map $x mapsto x'$, i.e. the identity on the underlying manifold $mathbb{R}$.