 Given two varieties \(V, V'\) in \(A^n\) and \(A^m\) respectively, a regular function from \(A^n\) to \(A^m\) whose restriction to \(V\) is a function from \(V\) to \(V'\) is called a regular function of varieties. The category of affine varieties is then defined to have as its objects all affine varieties and as its morphisms all regular functions thereof.