Let $\mathscr{P} : \textbf{Set} \to \textbf{Set}^\textrm{op}$ be the (contravariant) powerset functor, taking a set $X$ to its powerset $\mathscr{P}(X)$ and a map $f : X \to Y$ to the inverse image map $f^* : \mathscr{P}(Y) \to \mathscr{P}(X)$. By elementary abstract nonsense, we know that $\mathscr{P}$ has a right adjoint $\mathscr{P}^\textrm{op}$: indeed,
$\textrm{Hom}(X, \mathscr{P}(Y)) \cong \textrm{Hom}(X \times Y, \Omega) \cong \textrm{Hom} (Y \times X, \Omega) \cong \textrm{Hom}(Y, \mathscr{P}^\textrm{op} (X)) $
where $\Omega$ is the subobject classifier $\{ 0, 1 \}$ for $\textbf{Set}$. Let us write $\mathscr{T}$ for the composite $\mathscr{P}^\textrm{op} \mathscr{P}$; then we have a monad $(\mathscr{T}, \eta, \mu)$ on $\textbf{Set}$.
By advanced abstract nonsense (Paré's theorem or Mikkelsen's theorem) it can be shown that $\mathscr{P}$ is monadic, i.e. that the canonical comparison functor $\textbf{Set}^\textrm{op} \to \mathscr{T}\textbf{-Alg}$ is an equivalence of categories; but $\textbf{Set}^\textrm{op}$ is known to be equivalent to the category of complete atomic boolean algebras, so this therefore implies that a $\mathscr{T}$-algebra is the same thing as a complete atomic boolean algebra.
Question. How does one show this directly?
It is straightforward to check that $\eta_X (x) = \{ S \subseteq X : x \in S \} = \mathord{\uparrow} (\{ x \})$ and the multiplication map is $\mu_X (V) = \bigcup_{U \in V} \{ S \subseteq X : \eta_{\mathscr{P} (X)}(S) = U \}$ but I am having a lot of trouble getting any intuition for what $\mu_X$ does beyond how it is constructed: it recovers the set of all $S$ such that the principal filter $\mathord{\uparrow} (\{ S \})$ is a member of $V$. If $V$ itself is upwards-closed, then we have the formula $\mu_X (V) = \bigcup_{U \in V} \bigcap_{T \in U} T$ since for all $U \in V$, if $S \in T$ for all $T \in U$, we have $U \subseteq \mathord{\uparrow}(\{ S \})$, i.e. $U \subseteq \bigcap \left\{ \mathord{\uparrow}(\{ S \}) : S \in \bigcap U \right\}$ Thus, for $U \subseteq \mathscr{T}(X)$, $\mu_X ( \mathord{\uparrow}(U) ) = \bigcap_{T \in U} T$ and as a special case we obtain the left unit law $\mu_X ( \eta_{\mathscr{T}(X)} (T) ) = T$ This means that we can recover the meet operation of $\mathscr{T}(X)$ from $\mu_X$, regarded as the free complete atomic boolean algebra on $X$. We can also recover the join operation: $\bigcup_{T \in U} T = \mu_X \left( \bigcup \left\{ \mathord{\uparrow} (\{ T \}) : T \in U \right\} \right)$ (The use of $\bigcup$ in the RHS is legitimate since the join operation of $\mathscr{T}(\mathscr{T}(X))$ independent of the join operation of $\mathscr{T}(X)$.)
Now, I know that any complete lattice $A$ admits a canonical pseudocomplement operation \lnot a = \bigvee \{ a' \in A : a \wedge a' = \bot \} but I have no idea how the properties of $\mu_X$ imply the ‘meet’ and ‘join’ operations we get satisfy the requirements for a (complete) boolean algebra, nor do I see where atomicity might be coming from.