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In this paper, we construct the $\mathbb Z_{2}^{n}-$supergrassmannians bygluing of the $\mathbb Z_{2}^{n}-$superdomains and give an explicit descriptionof the action of the $\mathbb Z_{2}^{n}-$super Lie group$GL(\overrightarrow{\textbf{m}})$ on the $\mathbb Z_{2}^{n}-$supergrassmannian$G_{\overrightarrow{\textbf{k}}}(\overrightarrow{\textbf{m}})$ in the functorof points language. In particular, we give a concrete proof of the transitivelyof this action, and the gluing of the local charts of the supergrassmannian.

On the Construction of $\mathbb{Z}_2^n-$Supergrassmannians as Homogeneous $\mathbb{Z}_2^n-$Superspaces