Let $C$, $D$ be to categories. Now let $J$ be a collection of functors from $C$ to $D$. Then, what is the full subcategory of $Fun(C,D)$ spanned by J?
Basically, I am trying to understand the definition of a pro-object of a category. In one of Lurie’s notes, he defined the category of pro-objects of $C$, where $C$ is a category which has all finite limits, to be the opposite category of the full subcategory of $Fun(C,Set)$ spanned by functors which they preserve finite limits.