Definition:Embedding of Categories

Definition
Let $C$ and $D$ be categories.

Let $F : C \to D$ be a functor.

Definition 1
The functor $F$ is an embedding it is:
 * injective on objects
 * faithful

Definition 2
The functor $F$ is an embedding it is injective on morphisms.

Definition 3
The functor $F$ is an embedding it is a monomorphisms in the category of categories.

Also see

 * Equivalence of Definitions of Embedding of Categories