Presheaf category ($[{\mathcal{C}}^{\mathrm{op}},\mathbf{Set}]$)

The presheaf category on a small category $\mathcal{C}$ is the functor category $[{\mathcal{C}}^{\mathrm{op}},\mathbf{Set}]$. Its limits and colimits are computed pointwise.

Every presheaf category is a topos. Finite limits are pointwise; the exponential isand the subobject classifier assigns to $C$ the set of sieves on $C$.