I'm learning Sheaf Theory, and this is an issue that's been bothering me.

Fix a small category $\mathcal{C}$.

A $\mathcal{V}$-valued presheaf on the small category $\mathcal{C}$ is a functor $F:\mathcal{C}^{\text{op}}\rightarrow \mathcal{V}$. This determines a category $[\mathcal{C}^{\text{op}},\mathcal{V}]$ where objects are $\mathcal{V}$-valued presheaves, and morphisms are natural transformations between them.

If we impose certain gluing conditions (namely a Grothendieck Topology $J$), we can form the full subcategory $\text{Sh}_{\mathcal{V}}(\mathcal{C},J)$ of $[\mathcal{C}^{\text{op}},\mathcal{V}]$ called the category of $\mathcal{V}$-valued sheaves on the category $\mathcal{C}$.

There is also a dual notion called a copresheaf, which is a functor $F:\mathcal{C}\rightarrow \mathcal{V}$, determining a category $[\mathcal{C},\mathcal{V}]$ of copresheaves similar to earlier. Cosheaves form a category (which I'll denote $\textbf{CoSh}_{\mathcal{V}}(\mathcal{C},J)$) by a similar construction. (One can also view co(pre)sheaves $\mathcal{C}$ as (pre)sheaves on $\mathcal{C}^{\text{op}}$)

If $\mathcal{V}$ is an abelian category, then studying $\text{Sh}_{\mathcal{V}}(\mathcal{C},J)$ gives rise to Sheaf Cohomology. Also, if $\mathcal{V} = \textbf{Set}$, then $\text{Sh}_{\textbf{Set}}(\mathcal{C},J)$ is a very important example of a topos.

My question is why are (pre)sheaves more studied/important than co(pre)sheaves, when $\mathcal{C}$ and $\mathcal{C}^{\text{op}}$ are both small categories? Do (pre)sheaves give more important information about the category $\mathcal{C}$ (depending on what one is studying)?

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    $\begingroup$ A simple answer is that the Yoneda embedding $C \to [C^{op}, \text{Set}]$ is covariant. There is a sort of dual to the Yoneda embedding $C \to [C, \text{Set}]^{op}$ but $[C, \text{Set}]^{op}$ isn't a very nice category for various reasons, e.g. it's not locally presentable. $\endgroup$ – Qiaochu Yuan Mar 30 at 22:53

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