Comparison theorem over a base

Theorem 21.7.3 (Comparison over a base). Let

be a triangle equipped with a natural isomorphism \[ \theta \colon G'K \iso G \] Assume that \(G\) and \(G'\) admit left adjoints \(F\) and \(F'\), and that the mate \(\theta ^\flat \colon F' \to KF\) is an isomorphism. Assume furthermore that:

(1)

The functors \(G\) and \(G'\) are conservative;

(2)

The \(\infty \)-category \(D\) admits geometric realizations of \(G\)-split simplicial objects, and \(G\) preserves them;

(3)

The \(\infty \)-category \(D'\) admits geometric realizations of \(G'\)-split simplicial objects, and \(G'\) preserves them.

Then \(K\) is an equivalence of \(\infty \)-categories.