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 $Sh^\infty(\mathsf{Man})$
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> > let us write $f\sim g$ if there exists a (necessarily invertible) 2-morphism $f\Rightarrow g$ in $Sh^\infty(\mathsf{Man})$, and
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> > 
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> > 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 $h|\_{X\times\{1\}}\sim g$.
> > 
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> 


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?
[In the RHS of (\*), the symbol ∼ just means "homotopic" (and there's only one notion of two morphisms in $\mathcal S$ being homotopic)]


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?



Geometric realisation of smooth $\infty$-stacks
The case when $M$ is a smooth manifold follows from the [smooth Oka principle](https://ncatlab.org/nlab/show/shape+via+cohesive+path+%E2%88%9E-groupoid#ConsequenceSmoothOkaPrinciple).
See there for an expository account of the argument and references to additional sources.


Indeed, the left side of (\*) is $$\def\Hom{\mathop{\rm Hom}} π\_0(ʃ\Hom(M,X)),$$
whereas the right side of (\*) is $$π\_0(\Hom(ʃM,ʃX)),$$
where $ʃ$ denotes the [shape functor](https://ncatlab.org/nlab/show/shape+modality),
which is called “geometric realization” in the main post and is denoted by $‖{-}‖$ there.
(From my point of view, a geometric realization functor converts a categorical object like a simplicial set to a geometric object like a topological space, whereas a shape functor converts a geometric object like a sheaf of simplicial sets on manifolds to a categorical object like a simplicial set.)


Concerning the case of a general $M$,
not much can be expected if $M$ has homotopy groups in degree 1 or higher
(meaning $M$ is not weakly equivalent to a sheaf of sets on manifolds).
For example, taking $M=N/\!/G$ to be the stacky quotient of a manifold by an action of a Lie group, and $X$ to be the stack given by the homotopy group completion of the sheaf of symmetric monoidal groupoids of vector bundles,
the left side of (\*) computes the equivariant K-theory of $M$ (basically, it boils down to Segal's model),
whereas the right side computes the Borel equivariant K-theory of $M=N/\!/G$.
These two are different in general.


