Subcategory

A subcategory of a Category $\mathcal{C}$ is a subset of objects of $\mathcal{C}$, along with a subset of morphisms on those objects that is closed under composites and identities.

Alternatively, a subcategory of $\mathcal{C}$ is simply a Faithful Functor $\mathcal{C^{\prime}}\to\mathcal{C}$; this definition does not violate the Principle of Equivalence.