Let \(L\) be a modular complete lattice. The category \(\sigma[L]\) consists of all subgenerated lattices by a modular complete lattice \(L\). Here, the authors investigate the category \(\sigma[L]\). They relate the concept of a lattice subgenerated by a modular complete lattice \(L\) with the concept of the product of two lattices. They start with some definitions and results about lattices and describe the concepts of trace and generators. They give a new definition of trace related with the product in \(L\) of two lattices. In the second section, they investigate several properties of the category \(\sigma[L]\) such as self-generator and locally invariance in a lattice.