TITLE: Opposite category functor
QUESTION [5 upvotes]: I need some confirmation: is the opposite category transformation always a functor?
Also, isn't it always the case that $C^{\text{op}} = C$, since the the way we label an arrow does not matter?

REPLY [6 votes]: The objects of $\mathcal C^{\text{op}}$ are exactly the same as the objects of $\mathcal C$.
The morphisms of $\mathcal C^{\text{op}}$ are backwards versions of the ones in $\mathcal C$.
If we have a morphism $A \color{green}{\longrightarrow} B$ in $C$, we have a morphism $B \color{blue}{\longrightarrow} A$ in $\mathcal C^{\text{op}}$.
These are different categories because there might be a morphism $A \color{green}{\longrightarrow} B$ in $C$ but no morphism $A \color{blue}{\longrightarrow} B$ in $C^{\text{op}}$.
There is a contravariant functor $$ \begin{array}{rrcl} &\text{Dual}&:& \mathcal C \longrightarrow \mathcal C^{\text{op}} \\ &\text{Dual}(A)&=& A \\ &\text{Dual}(A \overset{f}{\color{green}\longrightarrow} B)&=& B \overset{f}{\color{blue}{\longrightarrow}} A \end{array}$$ which is an isomorphism of categories though.