There is definitely a category version of \$$\mathbf{Cat}\$$, where the objects are categories and the morphisms are functors. That's what we are using in this course - that's not a mistake, it's a useful thing.

There's also a 2-category version of \$$\mathbf{Cat}\$$, where the objects are categories, the morphisms are functors, and the 2-morphisms are natural transformations. This is ultimately more powerful, because natural transformation are very important - but we're not ready for it yet, because one has to learn categories before one can handle 2-categories.

(Math is like a game where there are always more 'levels': after we master one, we go on to the next.)