What is the opposite category of the category of Presheaves? I know that the category of presheaves on $\textbf{C}$ is defined as $\textbf{Psh}(C):= \textbf{Func}(\textbf{C}^{op}, \textbf{Set})$.
I have been wondering what the category $(\textbf{Psh}(\textbf{C}))^{op}$ would be like.
I have a hunch that $(\textbf{Psh}(\textbf{C}))^{op}$ would be isomorphic to $\textbf{Func}(\textbf{C}, \textbf{Set})$.
Am I correct in this guess? I have tried to work on this but I do not even know whether the hunch is correct. Any help with this would be great.
Thank you.
 A: This is not the case. For every category $\mathbf{Set}^{\mathbf{C}}$, its opposite category contains an object that is not the domain of any morphism except for its own identity (because the constantly $\varnothing$ functor is not the codomain of any morphism except its identity). But unless $\mathbf{C}$ is the empty category, $\mathbf{Set}^{\mathbf{C}}$ will never have such an object.
A: Presheaf categories are particularly nice examples of locally presentable categories, and you can show that the opposite of a locally presentable category is never locally presentable unless it is a preorder. So $\text{Psh}(C)^{op}$ is not isomorphic to any presheaf category. Many other arguments are possible; for example, presheaf categories are also Grothendieck topoi and so have, for example, exponential objects, while their opposites don't.
$\text{Psh}(C^{op})^{op}$ has a universal property dual to the universal property of presheaves: in the same way that $\text{Psh}(C)$ is the free cocompletion of $C$ (at least if $C$ is essentially small), $\text{Psh}(C^{op})^{op}$ is the free completion of $C$. That is, it's the category obtained from $C$ by "freely adjoining (small) limits." Mostly this is not too useful except as a convenient place to embed the category of pro-objects $\text{Pro}(C)$.
