Corollary 5.38.
For an acyclic class \(L\) in \(T\), the class \(D^{\infty}(L)\) is a congruence. Moreover, it is the largest congruence contained in \(L\). Thus \(L\mapsto D^{\infty}(L)\) defines a right adjoint \(\Acyc(T)\to\Cong(T)\) to the inclusion \(\Cong(T)\hookrightarrow\Acyc(T)\).
Proof
By Proposition 5.37, each \(D^n(L)\) is an acyclic class. Intersections of acyclic classes are acyclic, so \(D^{\infty}(L)=\bigcap_{n\geq0}D^n(L)\) is acyclic. It is closed under diagonals by construction and hence is a congruence by Proposition 5.35.If \(K\subseteq L\) is any congruence, then \(K\) is closed under diagonals. Inductively, every \(f\in K\) belongs to \(D^n(L)\) for all \(n\), so \(K\subseteq D^{\infty}(L)\). This proves maximality and the adjunction.