Expand RegexMatch To Test Against An Entire Range Of Cells - Google Sheets

In Google Sheets, I'm trying to essentially make an address book which I then want to use as a means of working out who is associated with each row in my data spreadsheet.

From looking around, I seemed to find that the best way to do the actual search would be to use the following formula

=ARRAYFORMULA(IF(REGEXMATCH($C$2:C, 'Address Book'!$A$2:A), 'Address Book'!$A$2:B ))

So my interpretation of this formula was that it is saying that for each row of column C, if it contains ANY of the values that are in Column A of the address book sheet, then show column A AND column B from the relevant row of the address book sheet.

However, I think the issue is actually that it is only looking at one row of the address book sheet as the formula only works if both the address book record and the data record use the same row number on their respective sheets. How can I expand this formula to make sure that it checks all of the values in column A of the address book sheet?