I am confused with morphisms of supermanifolds. Take a simple example $f:R^{0|1}\to R^{0|1}$. By (one of) definition, $f$ is a morphism of superalgebras of functions $C(R^{0|1})\to C(R^{0|1})$. Morphisms of superalgebras preserve the grading, I deduced that $f$ have the form $1\mapsto 1, \theta\mapsto x\theta$, i.e. $Hom(R^{0|1}\to R^{0|1})=R^1$ (as a set?). But I read from a paper that $Hom(R^{0|1},R^{0|1})=R^{1|1}$. What is going on? Thanks in advance!



Morphisms between supermanifolds R^{0|1}→R^{0|1}
You are right that the set of supermanifold morphisms 
$Hom(\mathbb R^{0|1},\mathbb R^{0|1})$ to itself is $\mathbb R^1$. 
However, one can define for supermanifolds $X,Y$ with $\dim X=0|d$ a 
supermanifold $map(X,Y)$ of morphisms from $X$ to $Y$, by $Hom(Z,map(X,Y))=Hom(Z\times X,Y)$ for all supermanifolds $Z$. 


And $map(\mathbb R^{0|1},\mathbb R^{0|1})=\mathbb R^{1|1}$.


