Not very elegant, but a simple solution is just to repeat the countif multiple times to see if the names appear in any of the columns. Not very feasible though if your worksheet expands to many workers.

The date here is just linked by columns (Col C of Sheet2 maps to Col I of 'Work Sheet') but to make it more dynamic you can add extra criterion to the search

Cell C6 in Sheet 2 (copy right and down for the whole table):
=IF( COUNTIFS('WORK SHEET'!I$8:I$16,"I", 'WORK SHEET'!$E$8:$E$16, $B6) + COUNTIFS('WORK SHEET'!I$8:I$16,"I", 'WORK SHEET'!$F$8:$F$16, $B6) + COUNTIFS('WORK SHEET'!I$8:I$16,"I", 'WORK SHEET'!$G$8:$G$16, $B6) + COUNTIFS('WORK SHEET'!I$8:I$16,"I", 'WORK SHEET'!$H$8:$H$16, $B6) > 0, "I","O")