Does the empty category exist?

Question

It seems to me that the definition of a category permits an empty category to exist.

The class of objects is the empty class and the arrows are morphisms between objects of the empty class (i.e., also empty).

All arrows commute is vacuously true.

All objects have an identity arrow is also vacuously true.

Answer 1

Yes, that's right. The empty category is the initial object in $\mathbf{Cat}$, the category of small categories.