Coherent six-functor formalisms: Pro vs Solid

In the classical theory for coherent sheaves, the only missing piece in the Grothendieck six-functor formalism picture is j_! for an open immersion j. Towards fixing this gap, classically Deligne proposed a construction of j_! by extending the sheaf class to pro sheaves, while Clausen-Scholze provided another solution by extending the sheaf class to solid modules. In this work, we prove that Deligne's construction coincides with the Clausen-Scholze construction via a natural functor, whose restriction to the full subcategory of Mittag-Leffler pro-systems is fully faithful.