The Quran

وَكَذٰلِكَ أَعثَرنا عَلَيهِم لِيَعلَموا أَنَّ وَعدَ اللَّهِ حَقٌّ وَأَنَّ السّاعَةَ لا رَيبَ فيها إِذ يَتَنازَعونَ بَينَهُم أَمرَهُم ۖ فَقالُوا ابنوا عَلَيهِم بُنيانًا ۖ رَبُّهُم أَعلَمُ بِهِم ۚ قالَ الَّذينَ غَلَبوا عَلىٰ أَمرِهِم لَنَتَّخِذَنَّ عَلَيهِم مَسجِدًا

Muhammad Asad

AND IN THIS way¹ have We drawn [people's] attention to their story,² so that they might know - whenever they debate among themselves as to what happened to those [Men of the Cave]³ - that God's promise [of resurrection] is true, and that there can be no doubt as to [the coming of] the Last Hour. And so, some [people] said: "Erect a building in their memory;⁴ God knows best what happened to them." Said they whose opinion prevailed in the end: "Indeed, we must surely raise a house of worship in their memory!"

I.e., by means of the legend which has grown up around the Men of the Cave and, more particularly, by means of the allegoric use which the Qur'an makes of this legend.
Lit., "given knowledge about them [to others]".
Lit., "debate their case (amrahum) among themselves": an indication of the fact that the legend of the Men of the Cave occupied men's minds for a long time, leading to many discussions and conflicting interpretations. The next sentence explains why God has "drawn [people's] attention" to this story in the context of the Qur'an.
This, to my mind, is the meaning of the expression 'alayhim (lit., "over them") occurring here as well as in the subsequent reference to the building of a house of worship at the suggestion of those "whose opinion prevailed in the end" (alladhina ghalabu 'ala amrihim).