Wa `alaykum As-Salamu wa Rahmatullahi wa Barakatuh

 

Tahajjud ends with the arrival of Fajr. Therefore, you should not deliberately postpone tahajjud until the fajr time arrives.

If, however, you started the prayer and the Fajr adhan starts, then you can still complete the prayer as long as the congregational prayer did not begin – assuming that you are praying in the mosque, in which case, you ought to join it.

 

Almighty Allah knows best.