>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.