  Let $\mathsf{s}_0\mathsf{Lie}^r$ be the category of $0$-reduced simplicial
restricted Lie algebras over a fixed perfect field of positive characteristic
$p$. We prove that there is a full subcategory
$\mathrm{Ho}(\mathsf{s}_0\mathsf{Lie}^r_{\xi})$ of the homotopy category
$\mathrm{Ho}(\mathsf{s}_0\mathsf{Lie}^r)$ and an equivalence
$\mathrm{Ho}(\mathsf{s}_0\mathsf{Lie}^r_{\xi})\simeq\mathrm{Ho}(\mathsf{s}_1\mathsf{CoAlg}^{tr})$.
Here $\mathsf{s}_1\mathsf{CoAlg}^{tr}$ is the category of $1$-reduced
simplicial truncated coalgebras; informally, a coaugmented cocommutative
coalgebra $C$ is truncated if $x^p=0$ for any $x$ from the augmentation ideal
of the dual algebra $C^*$. Moreover, we provide a sufficient and necessary
condition in terms of the homotopy groups $\pi_*(L_\bullet)$ for $L_\bullet \in
\mathrm{Ho}(\mathsf{s}_0\mathsf{Lie}^r)$ to lie in the full subcategory
$\mathrm{Ho}(\mathsf{s}_0\mathsf{Lie}^r_{\xi})$.
  As an application of the equivalence above, we construct and examine an
analog of the unstable Adams spectral sequence of A. K. Bousfield and D. Kan in
the category $\mathsf{s}\mathsf{Lie}^r$. We use this spectral sequence to
recompute the homotopy groups of a free simplicial restricted Lie algebra.
