7 · Follow the six operations

The six operations are tensor product, internal Hom, pullback, ordinary pushforward, compactly supported pushforward, and its right adjoint. They form three adjoint pairs.

Try this. Choose compactly supported pushforward for a proper map and then for a map with a boundary. In the proper case, it equals ordinary pushforward. Otherwise, it removes contributions that escape through the boundary.

Nagata compactification factors a separated finite-type map into an open immersion followed by a proper map. These rules define a candidate operation, while the theorem proves that it is independent of the chosen factorization.