A number of methods have been proposed in the recent years to perform the numerical homogenization of (possibly nonlinear) elliptic operators. These methods are usually defined at the discrete level. Most of them compute a numerical operator, close, in a sense to be made precise, to the homogenized elliptic operator for the problem. The purpose of the present work is to clarify the construction of this operator in the convex case by interpreting the method at the continuous level and to extend it to the nonconvex setting. The discretization of this new operator may be performed in several ways, recovering a variety of methods, such as the multiscale finite element method (MsFEM) or the heterogeneous multiscale method (HMM). In addition to the above, we introduce an original and general numerical corrector in the convex case.