If $F$ is a sheaf on a topological space $X$, the well-known étale space contruction gives rise to a bundle $\Gamma F$ on $X$ such that $F$ is isomorphic to the sheaf of sections of $\Gamma F$.

On the other hand, if $\mathfrak X$ is a stack on $C$ the Grothendieck contruction gives an equivalence between the category of sheaves on $\mathfrak X$ and the category of faithful morphisms of stacks on $C$ with target $\mathfrak X$.

My question is:

1: Is the Grothendieck construction a generalisation of the étale space construction?

In that case we could try to obtain the analogue result in the context of differentiables spaces or stacks:

2: If $\mathfrak X$ is now a differentiable space, is there an equivalence between the cat of sheaves on $\mathfrak X$ and the cat of faithful morphisms of differentiable spaces with target $\mathfrak X$?

3: Same question for $\mathfrak X$ a differentiable stack.

If $F$ is a sheaf on a topological space $X$, the well-known étalé space contruction gives rise to a bundle $\Gamma F$ on $X$ such that $F$ is isomorphic to the sheaf of sections of $\Gamma F$.

On the other hand, if $\mathfrak X$ is a stack on $C$ the Grothendieck contruction gives an equivalence between the category of sheaves on $\mathfrak X$ and the category of faithful morphisms of stacks on $C$ with target $\mathfrak X$.

My question is:

1: Is the Grothendieck construction a generalisation of the étalé space construction?

In that case we could try to obtain the analogue result in the context of differentiables spaces or stacks:

2: If $\mathfrak X$ is now a differentiable space, is there an equivalence between the cat of sheaves on $\mathfrak X$ and the cat of faithful morphisms of differentiable spaces with target $\mathfrak X$?

3: Same question for $\mathfrak X$ a differentiable stack.