Each one of these methods is approved and correct, but one may fit your subject better than the other one.
The introduction for a subject-by-subject type of compare and contrast essay should explain every information about the first subject and then moving on to the second one.
Depending on the amount of information you have about the subject, there are two ways to proceed with this method.
You can shortly describe the difference between the two subjects using only a paragraph for both of them or if you have plenty of time and information, you can devote a paragraph for only one subject.
For example: Tell us something we don't know (or might not notice)!
It would be better to write about how sensitive both horses and cats are to human needs and emotions.
With the opening paragraph, you must explain why the comparison between the two would be beneficial to the reader and to everyday life.
A strong introduction will help a lot with the grade and will catch even more attention than other essays.
If you are writing a lens compression, meaning that you are comparing the second subject solely based on the first one, you will want to use the subject-by-subject method because it makes writing easier.
For example, you might have an assignment to compare a book you have read in class and a book you have personally read at home.