Let $Sets$ be the category of finite sets and all maps. I have come accross several example of functors $F : Sets^{op} \to Sets$ which satisfy the condition below:

-- There exists an integer $k$ such that, whenever $X$ is a finite set, with subsets $U_1, U_2, U_3...$ whose union is $X$ and with $|U_i| \ge k$, then we have the usual glueing condition: given elements $a_i \in F(U_i)$ with $a_i|_{U_i\cap U_j} = a_j|_{U_i\cap U_j}$, there exists a unique $a \in F(X)$ with $a |_{U_i} = a_i$.

(I hope the meaning of the "restriction" notation is obvious.)

So far I have taken this, informally, to mean that $F$ "is determined by what it does to sets of size $\le k$" (whenever $|X| > k$, we can recover $F(X)$ from all the sets $F(U)$ where $U$ runs through the subsets of $X$ of size $k$). However, it would be instructive to have a more conceptual understanding of what's going on.

Obviously the condition is almost the sheaf condition, but of course, the "subsets of size $\ge k$" do not form a Grothendieck topology...

I know of one way to rephrase this: given a contravariant functor $G$ from $C_k$, the category of sets of size $\le k$, to $Sets$, you can extend it to a functor $R(G)$ on $Sets^{op}$ in such a way that $R$ is a right adjoint to the "truncation" functor, which takes $F$ to its restriction $F_{\le k}$ to $C_k^{op}$. And my condition is that $F$ is isomorphic to $R(F_{\le k})$ (I think this is called the $k$-th "coskeleton" of $F$). However, I don't really know what to make of this.

Does anyone know of a good conceptual framework for this situation ? Perhaps in terms of sheaves and Grothendieck topologies?

Edit : I like the answer by T; Goodwillie below.

If there is such a framework, then what can I learn about $F$, concretely?

Edit : let me more specific. For a given $F$, I'm trying to look at the minimal $k$ such that the above holds, and I'm hoping that $k$ is a good measure of the "complexity" of $F$, with a low value of $k$ meaning that $F$ is "easier" to understand. For example you get $k=1$ for $F(X) = Hom(X, S)$ for a fixed $S$; I think the converse holds, with $S=F(\{x\})$, which is defined up to a canonical bijection.

So I would like to

(1) Substantiate the hope that $k$ is a good quantity to look at. Any interpretation à la $n$-lab is good for this, and by now I'm convinced that it's reasonable. Extra comments are welcome of course.

(2) compute $k$ concretely in some examples. Which is why any observations about the consequences of the above condition may help put some bounds on $k$.