Let $\mathcal{C}$ be a category, and $Pro(\mathcal{C})$ the category of small cofiltered limits $I \to \mathcal{C}$. Dually, we have the category $Ind(\mathcal{C})$ of filtered colimits in $\mathcal{C}$, which may be thought of as cofinally small left exact presheaves on $\mathcal{C}$. Given that there is a canonical functor $Pro(\mathcal{C}) \to PSh(\mathcal{C})$, what is the corresponding characterization of pro-objects as presheaves? I know that $Pro(\mathcal{C})$ is equivalent to $Ind(\mathcal{C}^{op})^{op}$ but all the opposites are making me confused.


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