Definition:Subcategory

Definition 1
Let $\mathbf C$ be a metacategory.

A subcategory of $\mathbf C$ is a metacategory with:

Colloquially, a subcategory of $\mathbf C$ is a "part of $\mathbf C$ that is a category in its own right".

Definition 2
Let $\mathbf C$ be a metacategory.

A subcategory of $\mathbf C$ is a monic functor $F: \mathbf D \to \mathbf C$ to $\mathbf C$

Also see

 * Full Subcategory