This is just a sheaf valued in the ∞-category $\mathrm{Space}^{op}$. It is usually called a cosheaf. A place where this kind of thing shows up is in factorization algebras, that can be described as particular cosheaves over the Ran space.

I don't think they behave significantly differently from sheaves in any other complete ∞-category.

This is just a sheaf valued in the ∞-category $\mathrm{Space}^{op}$. It is usually called a cosheaf. A place where this kind of thing shows up is in factorization algebras, that can be described as particular cosheaves over the Ran space.

I don't think they behave significantly differently from sheaves in any other complete ∞-category, so I do not believe there are specific references for them.