Let $\mathcal{C}$ be a category without products, can we "complete" it into a category with products?
A presheaf is a contravariant functor from some topology to a category, in order to describe the condition of being a sheaf, we need $\mathcal{C}$ to have products. I am wondering if we can sheafify a general presheaf into a sheaf, but I am not sure how to define the concept of a sheaf in general case?
Consider the category of presheaves $[C^{op},Set]$ along with the yoneda embedding $Y : C \to [C^{op},Set]$. $[C^{op},Set]$ is closed under limits and colimits because $Set$ is, and they are computed pointwise. In particular, products exists.