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.

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.