For every stable presentably symmetric monoidal \(\infty \) -category \(\mathcal {C}\) and every non-unital \(\infty \) -operad \(\mathcal {O}\) in \(\mathcal {C}\) , where , we construct a Koszul duality adjunction \(\begin{aligned} {\textrm{TQ}}_\mathcal {O}: \textrm{Alg}_\mathcal {O}(\mathcal {C}) \leftrightarrows \textrm{Coalg}_{\mathcal {O}^\vee }(\mathcal {C}): {\textrm{Prim}}_\mathcal {O}\end{aligned}\) between \(\mathcal {O}\) -algebras in \(\mathcal {C}\) and coalgebras over the Koszul dual \(\infty \) -cooperad of \(\mathcal {O}\) . We prove that if all norm maps in \(\mathcal {C}\) associated to symmetric groups are equivalences, the unit of Koszul duality \(X \rightarrow {\textrm{Prim}}_\mathcal {O}({\textrm{TQ}}_\mathcal {O}(X))\) identifies with the canonical map \( X \rightarrow X^\wedge := \lim _{n \ge 1}\tau _n(\mathcal {O}) \circ _\mathcal {O}X \) to the limit of the \({\textrm{TQ}}_\mathcal {O}\) -completion tower \(\begin{aligned} X \simeq \mathcal {O}\circ _\mathcal {O}X \rightarrow ... \rightarrow \tau _n(\mathcal {O}) \circ _\mathcal {O}X \rightarrow ... \rightarrow \tau _1(\mathcal {O}) \circ _\mathcal {O}X= {\textrm{TQ}}_\mathcal {O}(X). \end{aligned}\) We apply this result to the Koszul duality between the shifted spectral Lie \(\infty \) -operad and the cocommutative cooperad to construct a derived enveloping Hopf algebra functor \(\textrm{Alg}_{\textrm{Lie}}(\mathcal {C}) \rightarrow {\textrm{Hopf}}(\mathcal {C}) \) from Lie algebras in \(\mathcal {C}\) to cocommutative Hopf algebras in \(\mathcal {C}\) and deduce a derived version of the Milnor-Moore theorem: for every rational stable presentably symmetric monoidal \(\infty \) -category \(\mathcal {C}\) the derived enveloping Hopf algebra functor \(\textrm{Alg}_{\textrm{Lie}}(\mathcal {C}) \rightarrow {\textrm{Hopf}}(\mathcal {C}) \) is fully faithful.
