On the definition of a subcategory

Question

When someone writes ''let $\mathcal{D}$ be a subcategory of a category $\mathcal{C}$'', is it possible that $\mathcal{C}=\mathcal{D}$? In other words, is a category a subcategory of itself?

Answer 1

In the normal fashion of category theory it is best to think of the arrows between things. In this case, subcategory of $\mathcal{C}$ is a faithful functor into $\mathcal{C}$ that's injective on objects, i.e. a monomorphism in $\mathbf{Cat}$. If you wanted to be more strict, you could require it to be an inclusion. Either way, in this case $\mathcal{C}$ is a subcategory of itself via the identity functor which is trivially faithful and injective/an inclusion.