Assuming that Library and Symphony are different sheets and that Last names are in column B:
I would just put a formula in the column next to names that says:
=IF(COUNTIF('Library'!B:B,B1)>0,B1,"")
Then drag the formula down as far as the list goes.

This will make it so that all names that are in the Library list will be displayed next to their name on the Symphony list. You can then create conditional formatting to make it so that any non-blank cell in the column with the formula is highlighted.