There's an easy way to settle if the category of \$$X\$$s is a \$$X\$$ itself.

Take \$$X\$$ to be sets. The question of whether \$$\mathbf{Set}\$$, the category of sets, is itself is a set (a set of all sets) gives [Cantor's paradox](https://ncatlab.org/nlab/show/Cantor%27s+paradox).