Given a functor $F : A \rightarrow B$, I managed to define the functor $F^{op} : A^{op} \rightarrow B^{op}$.
Now, I cannot see how to use it to define a functor from and to $\text{Cat}$ such that $C \longmapsto C^{op}$ and $F \longmapsto F^{op}$ for $C$ category and $F$ functor.
For instance, suppose that $\text{id}$ is the identity functor. Now we have to show: $\text{id}_c^{op} = \text{id}_{c^{op}}$. Do I proceed by case analysis?
I also have trouble when proving that $(F \circ G)^{op} = F^{op} \circ G^{op}$ because I arrive at something resembling a contra-variant functor.
This exercise is taken from Lecture Notes: Introduction to Categorical Logic