Is it better to wait for the Adhan after the time for prayer has begun in order to perform it, or to hasten to pray immediately once its time has begun?

The entry of the time is a condition for the validity of the prayer, and the proof for this is the Almighty's saying: (Indeed, prayer has been decreed upon the believers a [definite] time). The adhan (definition) is an announcement of the time, so if the muezzin adheres to it, prayer is performed after it. If he delays, the Muslim prays when he is certain that the time has entered. Women are permitted to delay beyond the beginning of the time.