For a small category, and its presheaf topos, we have that a colimit-preserving functor is equivalently itself a copresheaf.
If we replace in this statement presheaves with sheaves, we obtain the notion of cosheaf
Let be a site. A cosheaf on is a copresheaf
such that it takes covers to colimits: for each covering family in we have
Write for the full subcategory of cosheaves.
There is a natural equivalence of categories
where on the right we have the category of colimit-preserving functors.
Equivalently: a copresheaf is a cosheaf precisely if its Yoneda extension factors through sheafification .
This is (BungeFunk, prop. 1.4.3).
Cosheaves of algebras, or notions similar to this, appear in AQFT as local nets of observables. Similar structures in higher category theory are factorization algebras, factorization homology, topological chiral homology.
section 1.4 of
Marta Bunge and Jonathan Funk, Singular coverings of toposes Lecture Notes in Mathematics, (2006) Volume 1890/2006
chapter 1 Lawvere Distributions on Toposes