We prove that the dihedral Lie coalgebra $D\_{\bullet \bullet}:=\bigoplus\_{k\geq m \geq 1} D\_{m,k}$ corresponding to ${\widehat{\mathscr{D}}{\bullet \bullet}(G)}$ of Goncharov (Duke Math. J. 110 (2001), 397–487) for $G={e}$ is the bigraded dual of the linearized double shuffle Lie algebra $\mathfrak{ls}:=\bigoplus{k\geq m \geq 1}\mathfrak{ls}m^k\subset \mathbb{Q}\langle x,z \rangle$ of Brown (Compos. Math. 157 (2021), 529–572) whose Lie bracket is the Ihara bracket initially defined over $\mathbb{Q}\langle x,z \rangle$, by constructing an explicit isomorphism of bigraded Lie coalgebras $D{\bullet \bullet} \to \mathfrak{ls}^\vee$, where $\mathfrak{ls}^\vee$ is the Lie coalgebra dual (in the bigraded sense) to $\mathfrak{ls}$. The work leads to the equivalence between the two statements “$D\_{\bullet \bullet}$ is a Lie coalgebra with respect to Goncharov’s cobracket formula in Goncharov (Duke Math. J. 110 (2001), 397–487)” and “$\mathfrak{ls}$ is preserved by the Ihara bracket''. We also prove folklore results from Brown (Compos. Math. 157 (2021), 529--572) and Ihara et al. (Compos. Math. 142 (2006), 307--338) (which apparently have no written proofs in the literature) stating that for $m \geq 2$, $D\_{m,\bullet}:=\bigoplus\_{k\geq m} D\_{m,k}$ is graded isomorphic (dual) to the double shuffle space $\mathrm{Dsh}m:=\bigoplus{k\geq m} \mathrm{Dsh}{m}({{k}-m}) \subset \mathbb{Q}\[x\_1,\dots,x\_m]$ (stated in Ihara et al., Compos. Math. \textbf{142} (2006), 307–338), and that the linear map $f\_m\colon \mathbb{Q}\langle x,z \rangle\_m \to \mathbb{Q}\[x\_1,\dots,x\_m]$, where $\mathbb{Q}\langle x,z \rangle\_m$ is the space linearly generated by monomials of $\mathbb{Q}\langle x,z \rangle$ of degree $m$ with respect to $z$, given by $x^{n\_1}z\cdots x^{n\_m}zx^{n{m+1}}\mapsto \delta\_{0,n\_{m+1}} x\_1^{n\_1}\cdots x\_{n\_m}^{n\_m}$, with $\delta\_{a,b}$ the Kronecker delta, restricts to a graded isomorphism $\bar{f}m\colon \mathfrak{ls}m:=\bigoplus{k\geq m} \mathfrak{ls}m^k \to \mathrm{Dsh}{m}$ (stated in Brown (Compos. Math. 157 (2021), 529--572)). Here, we establish three explicit compatible isomorphisms $D{\bullet \bullet} \to \mathfrak{ls}^\vee, D\_{m\bullet}\to \mathrm{Dsh}{m}^\vee$ and $\bar{f}m\colon \mathfrak{ls}m \to \mathrm{Dsh}{m}$, where $\mathrm{Dsh}{m}^\vee$ is the graded dual of $\mathrm{Dsh}{m}$.