Full, Faithful and Embeddings

Given two categories $ \mathcal{C},\mathcal{D} $, and a functor $ F : \mathcal{C} \rightarrow \mathcal{D} $, we say that $ F $ is

full if for every two objects, $ A,B $ of $ \mathcal{C} $ we have that $$ F : \mathcal{C}(A,B) \rightarrow \mathcal{D}(F(A),F(B)) $$ is a surjection. We can make an example where this is not the case. Denote by subscript the category some object belong to, eg. $$ F(A_C) = A_D $$ Now say we have the arrows $$ f_1,f_2 : A_C \rightarrow B_C $$ in the category $ C $. But in the category $ D $ we have $$ g_1,g_2,g_3 : A_D \rightarrow B_D $$
faithfull if $ F $ is a always injective. We can again make a counter example. Say we in the category $ \mathcal{C} $ have the arrows $$ f_1,f_2,f_3 : A_C \rightarrow B_C $$ and in the category $ D $ we have the arrows $$ g_1,g_2 : A_D \rightarrow B_D $$ Here we let the functor $ F $ map as follows $$ [f_1 \mapsto g_1, f_2 \mapsto g_2,f_3 \mapsto g_2] $$

Remeber that $ \mathcal{C}(A,B) $ is the class of arrows in $ \mathcal{C} $ from the object $ A $ to the object $ B $.

Embeddings

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 $.