Six-Functor Formalisms III: The construction and extension of 6FFs

This article is the last of the series of articles where we reprove the foundational ideas of abstract six-functor formalisms developed by Liu-Zheng. We prove the theorem of partial adjoints, which is a simplicial technique of encoding various functors altogether by taking adjoints along specific directions. Combined with the $\infty$-categorical compactification theorem from the previous article, we can construct abstract six-functor formalisms in reasonable geometric setups of our interest. We also reprove the simplified versions of the DESCENT program due to Liu-Zheng, which allows us to extend such formalisms from smaller to larger geometric setups.