On a previous page we looked at ways to construct categories from existing categories. On this page we look at how objects in existing categories can become arrows in a new category. 
A specific case of arrow categories is comma categories and a more specific case is slice categories. We can then further generalise to pullbacks.
In some circumstances we will see that certain universal properties are conserved.
Arrow Category
The arrow category gives us a way to convert objects into arrows. 
objects:  f: A > X  Objects in this arrow category are arrows between two categories. To specify this completely we need a triple <A,X,f> consisting of the two objects and the morphism between them.  
morphisms:  <s,t>  where 's' is an arrow in the source and 't' is an arrow in the target. 
where the above diagram commutes, that is:
g•s = t •f
Comma Category
The comma category is a specialisation of the arrow category where the codomain is the same for all objects.  
Alternatively there is also a cocomma category where the domain is the same for all objects.  
The above diagram could be rewritten with the X shown in each object to make it look more like the arrow category. 
The concept of a comma category is related to the idea of a fibre and sheaf, reading these pages for may help with intuition.
The category of graphs is an example of a comma category, that is, a graph is isomorphic to a comma category. Arrows of graphs are pairs of functions mapping nodes to nodes (N > N) and edges to edges (E > E) , an object of this category it a triple (E,f: E > N×N,N) where f maps each edge to a source and target node.
Then the comma category CX has, 
The cocomma category XC has, 

Objects 

pairs(A,f)



Morphisms 




Terminal Objects in Comma 
Initial Objects in CoComma 

If C has a terminal object '1' exists then: C/1 = C 
If C has an initial object '0' exists then: 0/C = C 

Products in Comma 
CoProducts in CoComma 

In terms of the above 'pairs' we can construct a product:
We can reduce the above diagram to: 
Slice Category
This construction allows us to start with one category 'C' and generate a different category 'C/X' by fixing a given element 'X' in C. The elements in C/X are pairs (A,P) where AC and P is a morphism from A to X.
An example might be adding a distinguished point (the origin) to Euclidean space to give a vector space.
In the following we have also explained 'coslice' the dual concept in the yellow boxes on the right.
Given a category C we can 'slice' it over some object XC which we fix in C.
Comma category is a more general case of the slice category (slice category is a specific case of comma category). this time we fix a functor f from category C to category D with an object XD.
So we fix f: C>D and XD
Then the slice category C/X has, 
The coslice category X/C has, 

Objects 




Morphisms 



We can generalise this further by not fixing X
Instead x is combined with the pair to give a triple.
Then the comma category FX has, 
The cocomma category FC has, 

Objects 

triple(cC,xD,p)



Morphisms 


Further generalisation, we add a third category 'E' like this:
Then the comma category FX has, 
The cocomma category FC has, 

Objects 



Morphisms 

