>No, there is no such thing as a functor \\(\mathbf{op}: \mathcal{C} \to \mathcal{C}^{\mathrm{op}}\\). Functors can't 'flip arrows around', since a functor \\(F\\) maps a morphism \\(f : x \to y\\) to a morphism \\(F(f) : F(x) \to F(y)\\).

>This is a great example of a level slip!

Whoops! Especially after I chastised someone today for level slipping with relations...

Anyways, thanks for pointing that out.