  We study completion with respect to the iterated suspension functor on
$\mathcal{O}$-algebras, where $\mathcal{O}$ is a reduced operad in symmetric
spectra. This completion is the unit of a derived adjunction comparing
$\mathcal{O}$-algebras with coalgebras over the associated iterated
suspension-loop homotopical comonad via the iterated suspension functor. We
prove that this derived adjunction becomes a derived equivalence when
restricted to 0-connected $\mathcal{O}$-algebras and $r$-connected
$\tilde{\Sigma}^r \tilde{\Omega}^r$-coalgebras. We also consider the dual
picture, using iterated loops to build a cocompletion map from algebras over
the iterated loop-suspension homotopical monad to $\mathcal{O}$-algebras. This
is the counit of a derived adjunction, which we prove is a derived equivalence
when restricting to $r$-connected $\mathcal{O}$-algebras and $0$-connected
$\tilde{\Omega}^r \tilde{\Sigma}^r$-algebras.
