Array formula

Hello guys...

I've got a formula in my spreadsheet that is quite lengthy. I could use an array formula to replace this formula...but I don't know how to construct it.

Below is an example of my current formula:

=IF(AND($A1="Cats",$B1=1),"Cats 1",IF(AND($A1="Cats",$B1=2),"Cats 2",
IF(AND($A1="Dogs",$B1=3),"Dogs 1",IF(AND($A1="Dogs",$B1=4),"Dogs 2",
IF(AND($A1="Birds",$B1=5),"Birds 1",IF(AND($A1="Birds",$B1=6),"Birds 2",$A1))))))

The code in my spreadsheet is far longer than this example, as there are considerably more options for each category.

Attached is a sample workbook containing the example above.
The worksheet might better explain what I need the array formula to do.