I should put a little warning symbol here like John does. John said
> But a lot of modern category theorists, and certainly Fong and Spivak's book and me in this course, **never use** contravariant functors \$$F : \mathcal{X} \to \mathcal{Y} \$$. As a substitute, we always use ordinary functors \$$F : \mathcal{X}^{\text{op}} \to \mathcal{Y}\$$ or \$$F : \mathcal{X} \to \mathcal{Y}^{\text{op}} \$$.
>
> All three of these are just different ways of talking about the same idea.

Whilst it is true that the collection of functors \$$\mathcal{X}^{\text{op}} \to \mathcal{Y}\$$ is the same as the collection of functors \$$\mathcal{X}\to \mathcal{Y}^{\text{op}} \$$, these give rise to two different categories. Remember that the functor category \$$\operatorname{Fun}(\mathcal{A}, \mathcal{B})\$$ has functors from \$$\mathcal{A}\$$ to \$$\mathcal{B}\$$ as its objects and natural transformations between these functors as its morphisms. If you work it through, you will find that the morphisms in \$$\operatorname{Fun}( \mathcal{X}^{\text{op}} , \mathcal{Y})\$$ go 'the opposite way' to those of \$$\operatorname{Fun}( \mathcal{X}, \mathcal{Y}^{\text{op}} )\$$, so in fact

$\operatorname{Fun}( \mathcal{X}^{\text{op}} , \mathcal{Y})^{\text{op}} = \operatorname{Fun}( \mathcal{X}, \mathcal{Y}^{\text{op}}).$

So you should exercise a little care when talking of the *category* of contravariant functors.