can anyone explain why we cannot contrapose the second conditional relationship we made?

Any advice on reading difficult passages?

I am a little bit confused on one thing. When we negate both claims, the original necessary condition now becomes sufficient and the sufficient becomes necessary. Now what I don't understand is for this example, not being a cat is the new necessary condition so we are saying, 'not being a cat is necessary for not being a mammal' how does that make sense? it could be another animal like a dog for example.