A $\mathbb{C}$-motivic $w_1$-self-map of periodicity 1

Let $\mathrm{D}\mathrm{Y}=\mathbb{1}\mathrm{/}(\eta ,\nu )\in \mathrm{S}\mathrm{H}(\mathbb{C})$. We construct a $w_1$-self-map of periodicity 1 $w:{\mathrm{\Sigma }}^{5,3}\mathrm{D}\mathrm{Y}\to \mathrm{D}\mathrm{Y}\mathrm{.}$ The cofiber is a realization of $DA(1)$, the double of the subalgebra $A(1)$ of the $\mathbb{C}$-motivic Steenrod algebra $A$.