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?

Lac operon is a segment of $DNA$ that is made up of three adjacent structural genes, namely, an operator gene, a promoter gene, and a regulator gene. It works in a coordinated manner to metabolize lactose into glucose and galactose.

In $l a c$ operon, lactose acts as an inducer. It binds to the repressor and inactivates it. Once the lactose binds to the repressor, $RNA$ polymerase binds to the promoter region. Hence, three structural genes express their product and respective enzymes are produced. These enzymes act on lactose so that lactose is metabolized into glucose and galactose

. After sometime, when the level of inducer decreases as it is completely metabolized by enzymes, it causes synthesis of the repressor from regulator gene. The repressor binds to the operator gene and prevents $RNA$ polymerase from transcribing the operon. Hence, the transcription is stopped. This type of regulation is known as negative regulation. (figure)