Let $Sh^\infty(\mathsf{Man})$ denote the $\infty$-category of sheaves of $\infty$-groupoids over the site $\mathsf{Man}$ of smooth manifolds (if you prefer, that's the model category of simplicial sheaves on $\mathsf{Man}$), and let $\mathcal S$ denote the $\infty$-category of $\infty$-groupoids (the usual model category of simplicial sets).

The inclusion $\mathcal S\to Sh^\infty(\mathsf{Man})$ admits a left adjoint $$ \|\cdot\|:Sh^\infty(\mathsf{Man})\to\mathcal S $$ called geometric realisation.

Given two morphisms $f,g:X\to Y$ in $Sh^\infty(\mathsf{Man})$

let us write $f\sim g$ if there exists a (necessarily invertible) 2-morphism $f\Rightarrow g$ in $Sh^\infty(\mathsf{Man})$, and

let us write $f\approx g$ if there exists a map $h:X\times\mathbb R\to Y$ such that $h|_{X\times\{0\}}\sim f$ and $h|_{X\times\{1\}}\sim g$.

Is it true that for all $M\in\mathsf{Man}$, and all $X\in Sh^\infty(\mathsf{Man})$, the obvious map $$ \qquad\quad Hom_{Sh^\infty(\mathsf{Man})}(M,X)/\approx\quad \to \quad Hom_{\mathcal S}(\|M\|,\|X\|)/\sim\qquad\quad(*) $$ is bijective?

What can be said about the class of objects $M\in Sh^\infty(\mathsf{Man})$ with the property that $\forall X\in Sh^\infty(\mathsf{Man})$ the map $(*)$ is bijective?