As a next step up from a category of sets we can now look at the category of graphs. The objects in Grf consist of a set together with some internal structure.
Graphs also happen to represent category theory diagrams so perhaps we can find a way to relate the internal structure of graph objects to the category theory relationship of objects and arrows.
Category diagram for Graph
The category diagram for graph has two objects: a set of nodes and a set of edges. There are two arrows giving the source and target node for each edge. 
Functors Between Graphs
First lets look at the diagrams for graphs..
External diagram  Internal mapping for a specific example  

category diagram:  
simpler diagram which we can use in the case of endomaps: 
An endomap is a mapping from a object to itself. A category diagram for this case is shown in the higher row in the table above, we can see that it is an endomap because both the domain and the codomain are labled 'A', and hence are the same.
However, because this is an endomap, we can simplify the diagram by showing 'A' as one object.
Graph Categories
So lets start with Grf, a category of graphs, these have a bit more structure than sets and so will allow us to explore the relationships between objects which are sets+structure. I find it helpful to adapt the diagrams as bit, to get us started, so that we can include some of the normally hidden structure beneath. This is just to get us started, we won't always do this as the nature of category theory is to abstract as much as possible. Lets start with sets, which are perhaps the simplest objects, because they don't have the structure of operations like multiplications and additions.
Sets may just be any set of elements, or some or all of these sets may themselves be sets (and these inner sets may also contain sets and so on). Also these elements may all have a common type or not. So we could show the structure inside the set instead of representing them as single characters.  
In the case of structures like groups with a group operation then we could show that, perhaps by combining with a Cayley graph. 
Arrows
I find that including this further information helps even more with arrows:
Here I have showed individual arrows going from each element of the two objects rather than the more usual single arrow between the two objects.  
In the case of groups we may be able to give an indication of how the group operation is changed by the arrow. 
We will call the object at the start of the arrow the 'domain' and the object at the end of the arrow the 'codomain'
Subcategories of Grf
Two important subcategories are idempotent and invertible endomaps:
Idempotent Endomap All arrorw lead to fixed points. 

Invertible Endomap (permutation). contains only 2cycles and fixed points 
Irriflexive Graphs
Endomaps (mappings from an object to itself) are related to two parallel maps between different objects.
External diagram  Internal mapping for a specific example  

Irriflexive Graph:  
related endomap: 
The endomap is given by the difference between the red and blue mappings.
Adjunctions in Graphs
Adjunctions in Graphs
Adjunctions are described on page here.
Between reflexive graph and set. In reflexive graph: every node has loop to itself. 
Between irreflexive graph and set. In irreflexive graph: every node does not have loop to itself. 
Between dynamical system graph and set. In dynamical system graph: every node has one outgoing arrow. 
Different dynamical system graph and set. This time the morphism to set defines the fixpoints. 
Monoid
If we overlay multiple graphs (shown in different colours) then we get a Cayley graph as discussed on this page. This gives us a way to move onto monoids and groups in terms of category theory on this page.