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, 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.