In the medium where E. coli was growing, lactose was added, which induced the lac operon. Then, why does lac operon shut down some time after addition of lactose in the medium?

The lac operon consists of a regulatory gene and 3 structural genes. The lactose acts as an inducer. In the presence of an inducer, such as lactose, the repressor is inactivated by interaction with the inducer. This allows RNA polymerase access to the promoter and transcription proceeds. The repressor of the operon is synthesised. The repressor protein binds to the operator region of the operon and prevents RNA polymerase from transcribing the operon. Regulation of lac operon by repressor is referred to as negative regulation.