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

(N/A) The lac operon is a segment of $DNA$ consisting of three structural genes $(z, y, a)$, an operator gene, a promoter gene, and a regulator gene. It functions in a coordinated manner to metabolize lactose into glucose and galactose.
In the lac operon, lactose acts as an inducer. It binds to the repressor protein and inactivates it. Once the repressor is inactivated, $RNA$ polymerase can bind to the promoter region and initiate transcription. Consequently, the three structural genes express their products, and the respective enzymes ($\beta$-galactosidase, permease, and transacetylase) are produced.
These enzymes act on lactose, metabolizing it into glucose and galactose. After some time, as the level of the inducer (lactose) decreases due to its consumption by the enzymes, the repressor protein is no longer inactivated. The regulator gene synthesizes the active repressor, which binds to the operator gene and prevents $RNA$ polymerase from transcribing the operon. Thus, transcription is stopped. This type of regulation is known as negative regulation.