Here we look at building new categories from existing categories:
 Opposite Category
 Sub Objects
 Product (categorical product)
 Sum/Coproduct
 Exponent.
 Cartesian Product.
 Tensor Product.
 Functor Category.
 Comma/Slice Categories.
 Quotient Categories.
Opposite
There is a duality between categories by reversing the direction of the arrows.
The opposite category of C is denoted C^{op}
Sub Objects
This is a generalisation of subset.
The concept of a subobject is related to these concepts:
 Equivalence as described on page here.
 Subobject Classifier on page here.
 See page here for a noncategorical approach to subobjects.
Product and Sum
There are some 'universal constructions' such as product, sum and exponent that are objects that appear in many categories.
Product 
Sum 


universal cone over diagram  
generalisation  a kind of limit  a kind of colimit 
set example  cartesian product {a,b,c}*{x,y}= 
disjoint union {a,b,c}+{x,y}= 
group  the product is given by the cartesian product with multiplication defined componentwise.  free product the free product for groups is generated by the set of all letters from a similar "almost disjoint" union where no two elements from different sets are allowed to commute. 
Grp (abelian)  direct sum  direct sum the group generated by the "almost" disjoint union (disjoint union of all nonzero elements, together with a common zero) 
vector space  direct sum  direct sum 
poset  greatest lower bound meet 
least upper bound join 
base topological space  wedge  
POS  greatest lower bounds (meets) 
least upper bounds (joins) 
Rng  
Top  the space whose underlying set is the cartesian product and which carries the product topology  disjoint unions with their disjoint union topologies 
Grf  
category  objects: (a,b) morphism: (a,b)>(a',b') 
tensor products are not categorical products.
In the category of pointed spaces, fundamental in homotopy theory, the coproduct is the wedge sum (which amounts to joining a collection of spaces with base points at a common base point).
Sum
When generating a sum for objects with structure then the structure associated with the link can be added to the sum object.
Product
Products for groups are discussed on this page.
Cartesian Product
Here we extend the idea of a cartesian product of sets to other categories. The categorical product concept above defines the product by its external relationships which is the most categorical way to do it. Here we take a slightly less categorical approach and use the objects and arrows of the categories being multiplied to define their product. We treat the categories as set + structure.
The objects of the product are the same as the cartesian product of sets.
The arrows are combined in a similar way to the Cartesian and tensor products of graphs.
We can extend the concept of cartesian product of sets to categories as follows:
Here is an ordinary cartesian product of sets.  
Here we have sets with structure, represented by arrows. An arrow (a, b) > (a', b') exists in A×B if and only if either:


Tensor Product
More about this on this page.
We can extend the concept of cartesian product of sets to categories as follows:
The objects are the same as cartesian product of sets but the arrows are different from the cartesian example above. There is an arrow in AB only if there is an arrow in both A and B. An arrow (a, b) > (a', b') exists in AB if and only if:


Looking at the case above it looks like we could recover projections to A and B from AB. However, is there an example where we can't, where one of the operands has a object with no arrows leaving it? On the right is a diagram of an extreme case where one of the operands has no arrows. In this case, by the definition given above, the tensor product will have no arrows. In this case there may be a projection from A0 to 0 (that would be the Cartesian set product) Would there be an arrow from A0 to A? maybe so, if we take a graph with no arrows as being an initial object. Perhaps this is an example which shows the cartesian product of graphs is not a product in the category theory sense? 

A graph with a single object and an arrow to itself is the unit: 1. 
Exponential
This is a universal structure but not a limit.
Functor Category.
We can construct a new category from an existing category 'C', where:
 The objects are the functors of 'C'.
 The morphisms are the natural transformations of 'C'.
More about Functor Category on page here.
Example of Functor Category
The category of all directed graphs is the functor category Set^{C}. (see presheaves on page here)
Where C is a category with two objects connected by two morphisms
see Yoneda embedding.
Comma/Slice Categories.
More about Comma/Slice categories on this page.
Quotient Categories.