thing1 cat1 -> cat1
thing1 cat2 -> cat1
thing1 nocat -> cat1

thing2 cat1 -> cat1
thing2 cat2 -> cat1
thing2 nocat -> cat1

nothing cat1 -> cat1
nothing cat2 -> cat2
nothing nocat -> nocat

thing3 cat1 -> cat2
thing3 cat2 -> cat2
thing3 nocat -> cat2

thing4 cat1 -> cat2
thing4 cat2 -> cat2
thing4 nocat -> cat2