  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 $H_{Gr}^i\circ \textbf{exp}$ and $H^i_{Lie}$ are naturally
equivalent. A similar result is proven for $i=3$ and the relative cohomology
groups.
