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)?