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! What we really have is a functor

$\mathbf{op} : \mathbf{Cat} \to \mathbf{Cat}$

This functor sends each object of \$$\mathbf{Cat} \$$ - that is, each category \$$\mathcal{C}\$$ - to its opposite:

$\mathbf{op}(\mathcal{C}) = \mathcal{C}^{\mathrm{op}}$

This functor send each morphism of \$$\mathbf{Cat} \$$ - that is, each functor \$$F: \mathcal{C} \to \mathcal{D} \$$ - to some obvious thing.

**Puzzle.** What is this obvious thing?