This is the last talk towards understanding Bezrukavnikov-Finkelberg's derived geometric Satake equivalence. With the preparations from previous talks, we will introduce two filtrations: a topological filtration on the equivariant cohomology and an algebraic filtration on the Kostant functor. The comparison of these two functors will lead us to the main statement of Bezrukavnikov-Finkelberg's derived geometric Satake equivalence.