So if a horse in one sheet is listed in another sheet, then you want to say "Yes", otherwise leave it blank?
Looks like a vlookup may be your best bet, combined with an ISERROR something along the lines of:
=IF(ISERROR(VLOOKUP(A2,Sheet2!A:A,1,FALSE)),"","YES")
If your list of horses is:
Then this gives this output:
| Horse | |
| A | |
| B | YES |
| C | |
| D | |
| E | |
| F | |
| G | YES |
| H | YES |
| I | |
| J | |
| K | |
| L | |
Is that what you're after?