Transporting cohomology in Lazard correspondence

Lazard correspondence provides an isomorphism of categories between finitely generated nilpotent pro-$p$ groups of nilpotency class smaller than $p$ and finitely generated nilpotent ${\mathbb{Z}}_p$-Lie algebras of nilpotency class smaller than $p$. Denote by $H_{Gr}^i$ and $H_{Lie}^i$ the group cohomology functors and the Lie cohomology functors respectively. The aim of this paper is to show that for $i=0$, $1$ and $1$, and for a given category of modules the cohomology functors $H_{Gr}^i\circ \mathbf{\text{exp}}$ and $H_{Lie}^i$ are naturally equivalent. A similar result is proven for $i=3$ and the relative cohomology groups.