Original sentence: Several kinds of archaeological remnants of the Jerusalem Temple exist.There is extensive physical evidence for the temple called the Second Temple that was built by returning exiles around 516BCE and stood until its destruction by Rome in the year 70CE.Those for what is customarily called Solomon's Temple are indirect and some are challenged
Edited sentence: Several kinds of archaeological remnants of the Jerusalem Temple exist, both for the Temple that stood before the destruction of Jerusalem by Babylonia in 586 BCE, and for the rebuilt Temple that stood until  destruction by Rome in the year 70