Let $X$ be a spectrum. I've heard that the action of the Steenrod algebra on $H^*(X; \mathbb{F}_2)$ and the action of the Dyer-Lashof algebra on the homology of the associated infinite loop space, $H_*(\Omega^{\infty}X; \mathbb{F}_2)$, are "dual" in some sense.
To my knowledge, this has something to do with Koszul duality, but I've never seen this spelled out in detail. Anyone willing to enlighten me on this? Or point me toward a helpful reference?