Let $\mathsf{C}$ denote one of the categories $\mathsf{Set}$, $\mathsf{Group}$, $\mathsf{Ring}$, or $\mathsf{Mod}_R$, or some other sensible category in which we would like sheaves to take their values. Let $\text{Sh}(X, \mathsf{C})$ denote the category of $\mathsf{C}$-valued sheaves on a space (or more generally a site) $X$.
The initial object in $\text{Sh}(X, \mathsf{C})$ is obviously the constant sheaf $0_X$ corresponding to the initial object $0$ in $\mathsf{C}$, since the constant sheaf functor is left adjoint to the global sections functor.
But what is the terminal object? I would conjecture that it should be $1_X$, where $1$ is the terminal object in $\mathsf{C}$, but I do not see how to prove it.
Best Answer
The terminal presheaf is the constant presheaf with value the terminal object in $C$, since limits and colimits of presheaves are computed pointwise. The inclusion of sheaves into presheaves is a right adjoint, so preserves limits; hence the terminal presheaf is also the terminal sheaf.