Co-domain of a sheaf

Question

We know that sheaves take open sets in some topology to values in some category. I want to know if it's possible for a sheaf $\mathcal{F}$ to take values in some category $\mathcal{C}$ which has no products. I am asking this because I want to do the standard problem about the gluability of sheaves. It seems to me that it is just about taking products.

Answer 1

The definition of a sheaf works in any category. A functor $F : \mathsf{Open}(X)^{op} \to \mathcal{C}$ is a sheaf iff for every "test" object $T \in \mathcal{C}$ the functor $\hom(T,F(-)) : \mathsf{Open}(X)^{op} \to \mathsf{Set}$ is a sheaf in the classical sense.

In order to glue $\mathcal{C}$-valued sheaves, we need limits in $\mathcal{C}$.