Let $C$ be a site, $\mathbf{S}$ some ($\infty$-? homotopy?) category of spaces.

Question.What do you call a (covariant!) functor $F:C\to \mathbf{S}$ enjoying the following property: for every hypercovering $a\colon V_\bullet \to U$ in $C$, the induced map $$ {\rm hocolim}\, F(V_\bullet)\longrightarrow F(U) $$ is a homotopy equivalence?

Have such things been studied? References are welcome.