Definition of Supermanifold

Question

defining a super-domain as a pair $(U\subset\mathbb{R}^n, C^\infty(U)\otimes\bigwedge[\theta^1\cdots\theta^m])$, a supermanifold is usually defined as a topological space $M$ endowed with a sheaf of super-algebras (call it $C^\infty(\mathcal{M})$) which is locally isomorphic to a superdomain. My question is: how can I formulate this local isomorphism here? In one side, there is a given sheaf constructed in a topological space, in the other side there is an object constructed in the euclidean space.

Answer 1

The local isomorphism is a local isomorphism of ringed spaces, i.e. a homeomorphism $f: U\to V\subset M$ where $V$ and $U$ are open and an isomorphism of sheaves $F:f_*(C^\infty(\mathbb{R}^n)\otimes \bigwedge[\theta^1,\cdots, \theta^m])\to C^\infty(\mathcal{M})\vert_{V}$.

Edit: (I mixed up pullback and pushforward, resulting in something which did not make sense)