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I have long been confused about this notion. I know that for a statement within a single category, forming the dual statement is just reversing every arrows. But what about a statement concerning several categories and functors between them?

Mac Lane suggests reversing all arrows in all categories and leaving the functors invariant in his book "Categories for the working mathematician" (page 32), but it seems that Mac Lane himself doesn't always follow this discipline (when he states "the dual of Yoneda Theorem", he doesn't reverse the arrows in $\mathbf{Set}$ !) Can anyone give me a formal definition of "dualize a statement in category theory"?

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Some of the categories in a statement are "variables" (e.g. "$C$"), and you should take the opposite of those categories but not any categories which are "constants" (e.g. $\text{Set}$). The point is that any category which is a "variable" is being quantified over, so you can always replace that category with an opposite category, in the same way that you can substitute anything you want for a variable in an identity.

For example, if a functor $F : C \to D$ is a left adjoint, then it preserves colimits. The dual of this statement is obtained by replacing $C$ and $D$ with opposites, since they are both "variables," and you get that if $F : C^{op} \to D^{op}$ is a left adjoint, then it preserves colimits. But this is equivalent to a statement about $C$ and $D$ themselves, which is that if $F^{op} : C \to D$ is a right adjoint, then it preserves limits.

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    $\begingroup$ @Musa: no. Here you're only reversing arrows in categories. There just isn't a functor that goes in that direction. More formally, taking opposite categories is a covariant automorphism of the category of categories. $\endgroup$ – Qiaochu Yuan Feb 16 '16 at 21:38
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    $\begingroup$ @Censi: it depends on whether the theorem quantifies over all choices of $D$ or not (which is what I actually mean by "variable"), or at least quantifies over a collection of categories that are closed under taking opposites. For example, consider the statement "a category $C$ is closed under all finite colimits of diagrams $F : J \to C$ iff it is closed under finite coproducts and coequalizers." The dual statement is obtained by replacing both $J$ and $C$ with opposites, which we can do because finite categories are closed under opposites, and we get the dual statement... $\endgroup$ – Qiaochu Yuan Feb 17 '16 at 0:48
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    $\begingroup$ ...which is that a category is closed under all finite limits iff it is closed under finite products and equalizers. $\endgroup$ – Qiaochu Yuan Feb 17 '16 at 0:50
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    $\begingroup$ This condition about quantifying over a collection of categories closed under taking opposites is the real key here, and depending on what extra conditions you require on your categories it may or may not be available. For example, abelian categories are closed under opposites, but cartesian closed categories are not. $\endgroup$ – Qiaochu Yuan Feb 17 '16 at 4:34
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    $\begingroup$ @Censi: that statement is self-dual. The reason is that taking opposites is a covariant automorphism of Cat - it doesn't reverse functors - so products are still products. However, taking opposites reverses natural transformations, which is why left and right adjoints switch, and which will turn some 2-limits and 2-colimits into other 2-limits and 2-colimits. $\endgroup$ – Qiaochu Yuan Feb 17 '16 at 15:41

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