In wiki (link), an opposite category or dual category of of a given category C is formed by reversing the morphisms. It means the objects of are just the objects of C and the arrows of are arrows in a one to one correspondence manner (if an arrow of C is , then in is . I assume you already know this one.

Now consider some examples. Let S be a functor from the category of to category of B such that . It assigns to each object an object Sc of B and to each arrow of an arrow of B.

Here is another example. If is a functor from the category of C to the category of B, then we can define a functor from . In this case, can be applied to both, where c is an object of both C and C^{op} (Remember that objects remain same in the dual category). However, should be changed to for the latter one.