A semisimple mod $p$ Langlands correspondence in families for $GL_2(\mathbb{Q}_p)$

This is the sequel to arXiv:2007.01364v1. Let $F$ be any local field with residue characteristic $p>0$, and $\mathcal{H}^{(1)}_{\overline{\mathbb{F}}_p}$ be the mod $p$ pro-$p$-Iwahori Hecke algebra of $\mathbf{GL_2}(F)$. In arXiv:2007.01364v1 we have constructed a parametrization of the $\mathcal{H}^{(1)}_{\overline{\mathbb{F}}_p}$-modules by certain $\widehat{\mathbf{GL_2}}(\overline{\mathbb{F}}_p)$-Satake parameters, together with an antispherical family of $\mathcal{H}^{(1)}_{\overline{\mathbb{F}}_p}$-modules. Here we let $F=\mathbb{Q}_p$ (and $p\geq 5$) and construct a morphism from $\widehat{\mathbf{GL_2}}(\overline{\mathbb{F}}_p)$-Satake parameters to $\widehat{\mathbf{GL_2}}(\overline{\mathbb{F}}_p)$-Langlands parameters. As a result, we get a version in families of Breuil's semisimple mod $p$ Langlands correspondence for $\mathbf{GL_2}(\mathbb{Q}_p)$ and of Pa\v{s}k\={u}nas' parametrization of blocks of the category of mod $p$ locally admissible smooth representations of $\mathbf{GL_2}(\mathbb{Q}_p)$ having a central character. The formulation of these results is possible thanks to the Emerton-Gee moduli space of semisimple $\widehat{\mathbf{GL_2}}(\overline{\mathbb{F}}_p)$-representations of the Galois group ${\rm Gal}(\overline{\mathbb{Q}}_p/ \mathbb{Q}_p)$.