Given two categories $\mathcal{C},\mathcal{D}$, and a functor $F : \mathcal{C} \rightarrow \mathcal{D}$, we say that $F$ is
Remeber that $\mathcal{C}(A,B)$ is the class of arrows in $\mathcal{C}$ from the object $A$ to the object $B$.
If $F$ is full and faithful and injective on objects, we say the $F$ is an embedding. An embedding does not mean that the categories are equal up to isomorphism. With an embedding we are still allowed to have objects in the target category $\mathcal{D}$ that are not present in the source category $\mathcal{C}$. For example let $Cat_1$ be the category with objects $A$ and $B$ and arrows besides the $id$-arrows given as $$ f_1,f_2 : A \rightarrow B $$ Let $Cat_2$ be the category with objects $H$, $I$, $J$ and $K$ and arrows $$ g_1,g_2 : H \rightarrow I $$ along $$ g_3,g_4 : J \rightarrow K $$ Let $F : Cat_1 \rightarrow Cat_2$ the functor that maps objects as follows $$ [A \mapsto H,B \mapsto I] $$ and arrows as $$ [f_1 \mapsto g_1, f_2 \mapsto g_2] $$ Here $F$ is full, faithful and injective on objects. Hence $F$ is an embedding. It makes quite good intuitive sense that $Cat_1$ is embedded in $Cat_2$.