Given a Lie groupoid $G$, we can view it as representing a prestack on $\text{Mfld}$ by sending and manfold $M$ to the groupoid of smooth functors and smooth natural transformations
$$G(M) := \text{Fun}(M,G)$$
(viewing $M$ as a discrete groupoid). One of the multiple ways of improving this prestack to a stack is to instead use anafunctors and ananatural transformations. That is, we localize at coverings so that a manifold $M$ is sent to a groupoid containing objects of the form $M \leftarrow U \rightarrow G $ for coverings $U$ of $M$.


On the other hand, we can apply the plus construction to $G$ to get a stack. This is defined by
$$G^+(M) := \text{colim}\_{R \hookrightarrow M}\text{Psh}(R,G).$$
Here the colimit is taken over covering sieves of $M$. This seems to be doing the same thing as the anafunctors are. However, one would not expect to get a stack until the plus construction had been applied 3 times to $G$.


So, my question is: assuming these constructions really are doing the same thing, what about Lie groupoids lets us get away with only applying the plus construction once? Is this a general phenomenon with internal groupoids/geometric stacks, or is ther something special about the underlying site here?



Anafunctors vs the plus construction
The long-expected answer. $\DeclareMathOperator{\op}{op} \DeclareMathOperator{\Cat}{\mathbf{Cat}}\DeclareMathOperator{\Gpd}{\mathbf{Gpd}} \DeclareMathOperator{\disc}{disc}\DeclareMathOperator{\pr}{pr}$


Suppose for simplicity, we are in a site $(S,J)$ equipped with a subcanonical singleton pretopology $J$. This means every covering family consists of a single map that is an [effective epimorphism](https://ncatlab.org/nlab/show/effective+epimorphism) (the kernel pair exists because of the axioms of a pretopology, and the subcanonicity implies the map is a coequaliser of its kernel pair). For the purposes of your question, the category of smooth manifolds with surjective submersions as covers is such a site. Another way to get such a site is to consider an arbitrary subcanonical and *[superextensive](https://ncatlab.org/nlab/show/superextensive+site)* pretopology, then form the singleton pretopology consisting of the coproducts of the maps in the covering family. I'll come back to this later.


Recall that the definition of a prestack (for example Vistoli's *[Notes on Grothendieck topologies, fibered categories and descent theory](https://arxiv.org/abs/math/0412512)* Definition 4.6(i)) on an arbitrary site $(S,J)$ is a presheaf $F\colon S^{\op} \to \Cat$ such that for each object $M$ and covering family $\mathcal{U}$ of $M$ the comparison functor
$$
F(M) \to Desc\_F(\mathcal{U})
$$
to the category of [descent data](https://stacks.math.columbia.edu/tag/02ZC) is fully faithful. Let us spell out the definition of $Desc\_F(\mathcal{U})$ in the case that we have an internal groupoid $G$, and $F(M) = よ(G)(M) := \Gpd(S)(\disc(M),G)$, for $\disc$ denoting the internal groupoid with no non-identity arrows, and we have a singleton cover $j\colon U\to M$ in the pretopology $J$ as above.


An object of this category (following [Tag 026B](https://stacks.math.columbia.edu/tag/026B).(1)) is a internal functor $f\_0\colon \disc(U)\to G$ (equivalently, an arbitrary arrow $U\to G\_0$ of $S$, for $G\_0$ the objects of $G$), equipped with a natural isomorphism
$$
f\_1\colon f\_0\circ \pr\_1 \Rightarrow f\_0\circ \pr\_2,
$$
where $\pr\_i\colon U\times\_M U\to U$ projects on the $i^{th}$ coordinate. Such a natural iso is given by the data of an arrow $f\_1\colon U\times\_M U \to G\_1$ such that $(s\circ f\_1,t\circ f\_1) = (f\_0\circ \pr\_1,f\_0\circ \pr\_2)$. We also require that $f\_1$ satisfies as associativity constraint:
$$
\pr\_{23}^\*f\_1 \cdot \pr\_{12}^\*f\_1 = \pr\_{13}^\*f\_1\colon f\_0 \circ\pr\_1\Rightarrow f\_0\circ \pr\_3
$$
between these two functors $\disc(U\times\_M U\times\_M U) \to G$. Here $\cdot$ denotes the pointwise *internal* composition in $G$. This makes $f\_0$ and $f\_1$ the data of an internal functor from the Čech groupoid $\check{C}(U\to M)$ of the given cover, to $G$.


However, such a thing is precisely the data of an anafunctor $(U,f)\colon \disc(M)\leftarrow \check{C}(U\to M) \to G$! So we can identify the objects of the descent category with anafunctors, but this is not perhaps as helpful as it seems.


Next, let us calculate what the morphisms $(U,f)\to (U,g)$ are in the category (really, groupoid) of descent data (following [Tag 026B](https://stacks.math.columbia.edu/tag/026B).(2)). The data is a natural transformation $a\colon f\_0\Rightarrow g\_0\colon \disc(U)\to G$ (again, equivalently, a morphism $U\to G\_1$ in $S$ etc) satisfying the constraint that
$$
\pr\_2^\*a \cdot f\_1 = g\_1\cdot \pr\_1^\*a\colon f\_0\circ\pr\_1 \to g\_0\circ\pr\_2.
$$
This is precisely the definition of internal natural transformation between the two functors $f,g\colon \check{C}(U\to M)\to G$. Now this is *not* the same thing as a transformation between the anafunctors $(U,f)$ and $(U,g)$, the data of which would be a natural transformation $f\circ \pr\_1 \Rightarrow g\circ \pr\_2\colon \check{C}(U\times\_M U\to M) \to \check{C}(U\to M) \to G$.


Composition in this category of descent data agrees with the usual way of composing natural transformations, so that $Desc\_{よ(G)}(U\to M) = \Gpd(S)(\check{C}(U\to M),G)$. Moreover, the comparison functor $よ(G)(M) \to Desc\_{よ(G)}(U\to M)$ agrees with the precomposition functor $j^\*\colon \Gpd(S)(\disc(M),G) \to \Gpd(S)(\check{C}(U\to M),G)$.


So to show that $よ(G)$ is a prestack, we need to consider the action of the functor $j^\*$ on hom-sets. Fix then two morphisms $h,k\colon M \to G\_0$ of $S$ ($\Leftrightarrow$ functors $\disc(M)\to G$), and a natural transformation
$a\colon h\circ j \Rightarrow k\circ j$. First, note that the arrow component $(h\circ j)\_1\colon U\times\_M U \to G\_1$ is just $e\circ h \circ j'$ for $e\colon G\_0\to G\_1$ the unit map, and $j'\colon U\times\_M U\to M$ the projection. Similarly, $(k\circ j)\_1 = e\circ k \circ j'$.


The natural transformation is $a$ is given by the data of a function (abusing notation slightly) $a\colon U\to G\_1$ such that $\pr\_2^\*a \cdot (h\circ j)\_1 = (k\circ j)\_1\cdot \pr\_1^\*a$. Recall that here $\circ$ is composition of arrows in $S$, and $\cdot$ is pointwise composition of internal morphisms in $G$. Keeping this in mind, and using the calculation earlier, this gives
$$
\pr\_2^\*a \cdot (e\circ h \circ j') = (e\circ k \circ j')\cdot \pr\_1^\*a
$$
as natural transformations between functors $\check{C}(U\times\_M U)\to G$. But this amounts to saying two particular arrows $U\times\_M U \to G\_1$ are equal. What are these arrows?
Unwinding the above equation, each side is the internal composition of the component of a natural transformation with an identity arrow (because of the unit map $e$). But we need to calculate this in a slightly easier way.


We are now going to reason using generalised elements to check the two arrows $U\times\_M U \to G\_1$ are in fact equal.
Suppose that $(u\_1,u\_2)$ is a generalised element of $U\times\_M U$, satisfying $j(u\_1) = j(u\_2) =:m$. Then $\pr\_2^\*a(u\_1,u\_2) = a(u\_2)$ and $\pr\_1^\*a(u\_1,u\_2) = a(u\_1)$, generalised elements of $G\_1$. Now $a(u\_2) \colon h(j(u\_2)) \to k(j(u\_2))$, which is just $a(u\_2) = h(m) \to k(m)$ and likewise $a(u\_1) \colon h(m)=h(j(u\_1)) \to k(j(u\_1)) = k(m)$, as arrows in $G$. So $a(u\_1)$ and $a(u\_2)$ are parallel arrows. Then the condition they need to satisfy is nothing other than $a(u\_2)\cdot e\_{h(m)} = e\_{k(m)}\cdot a(u\_1)$, which is to say that $a(u\_2) = a(u\_1)$. This is true for arbitrary generalised elements, so by Yoneda we have $a\circ \pr\_2 = a\circ\pr\_1$, so that $a\colon U\to G\_1$ coequalises the parallel pair $\pr\_1,\pr\_2\colon U\times\_M U\rightrightarrows U$. But since $j$ is the coequaliser of this parallel pair, by subcanonicity of $J$, we have a *unique* $a'\colon M\to G\_1$ such that $a = j^\*a'$.


Thus $j^\*$ is fully faithful, and hence $よ(G)$ is a prestack.


Now, what if we don't want to use the singleton pretopology, but a general subcanonical superextensive pretopology? For instance, consider the case of the site of manifolds with open covers for covering families, or more generally, collections of submersions that are jointly-surjective. This is very close to, but not identical with, the site with only singleton covering families. In particular, one needs to consider the covering families given by coproduct injections, and of course everything generated from the combination of the two types of covering families. But, and this is the good point, an internal functor $\check{C}(\coprod\_{i\in I}U\_i\to M)\to G$ (given the extensivity of the ambient category $S$) is still equivalent to an object of the category of descent data! This is because a map $\coprod\_{i\in I}U\_i \to G\_0$ is equivalent to a collection of maps $\{U\_i \to G\_0\}\_{i\in I}$, and the definition of the descent category's objects in the non-singleton case amounts to defining a functor as above, and vice-versa. Similarly, the data of the natural transformation giving a morphism of descent data in the singleton case is equivalent to what it should be in the non-singleton case. The rest of the calculation goes through as before. One might also consider $\kappa$-extensivity for any given arity class $\kappa$ (eg finite extensivity, small extensivity, or even the degenerate case where we only consider "1-extensivity", which amounts to the original singleton case), and this all works fine.


So, to sum up: for an arbitrary subcanonical superextensive site $(S,J)$, the presheaf $よ(G)$ is a prestack for *any* internal groupoid $G$ in $S$ (so, for instance, one might wish to put some constraints of the source and target maps of $G$, but this is irrelevant to the above calculation). Note in particular that I haven't assumed $S$ has finite limits.


After this, one might want to consider the non-superextensive case. I believe that careful book-keeping should allow the above to go through, keeping track of the various components. In the end, one will use that the representable presheaf associated to the object $G\_1$ of $S$ is actually a sheaf, and the last calculation given checks that one has descent data for that sheaf. So I'm willing to claim one only needs $(S,J)$ subcanonical for this to work, and that the data of a topology is fine to use, rather than a pretopology. And, further, that really one only needs to know that the presheaf $よ(G\_1)$ (of sets) is a sheaf for the (pre)topology to get that the presheaf $よ(G)$ (of groupoids) is a prestack.


The only other generalisation I haven't touched on is when $G$ is instead an internal *category* that's not a groupoid. I believe this case should work too, since there was nothing in the above that actually *used* the fact $G$ was a groupoid. The category of descent data is then a category that's not a groupoid, but that doesn't affect anything, and I think the proof actually goes through unchanged. So, barring some tedious checking of book-keeping, I think I can claim:


**Theorem:** Given a site $(S,J)$ and an internal category $C$ in $S$ such that the representable presheaf on $S$ associated to the object of arrows $C\_1$ is a $J$-sheaf, the presheaf $よ(C)$ of categories is a prestack.


Now, to go back to the original question, this means that one only needs to apply the Grothendieck $+$-construction *once* to $よ(C)$ in order to get a stack on $(S,J)$.


Moreover, at least under the assumption that $(S,J)$ is subcanonical, and $J$ is singleton (or at the very least, superextensive), the [bicategory of internal anafunctors construction](http://www.tac.mta.ca/tac/volumes/26/29/26-29abs.html) gives a localisation of the 2-category $\Cat(S)$ (and analogously for the full sub-2-category $\Gpd(S)$) at the internal functors $\check{C}(U\to M) \to \disc(M)$ (using the more general version in *[The elementary construction of formal anafunctors](https://doi.org/10.52547/cgasa.15.1.183)*, using Example 2.11). ~~And we also have that the 2-category of stacks is *also* a localisation (from Pronk and Warren, *[Bicategorical fibration structures and stacks](http://www.tac.mta.ca/tac/volumes/29/29/29-29abs.html)*), and so the anafunctor bicategory and the 2-category of stacks are **equivalent** (here recalling we are in a subcanonical site, which Pronk and Warren assume). As a result objects of the the anafunctor bicategory essentially arise from performing the Grothendieck $+$-construction just once.~~


**EDIT** I need to correct that last statement. What is true is that the bicategory of internal categories, anafunctors and transformations is a localisation of the 2-category of internal categories, functors and transformations, and so is the 2-category of *presentable* stacks. Pronk and Warren's result is about localising the 2-category of prestacks (which the representable ones of the form $よ(G)$ form a full sub-2-category) at the local equivalences to get the 2-category of stacks. The local equivalences in the case of representable prestacks amount to (IIRC) the internal weak equivalences as in my work. The essential image of the 2-category of representable prestacks in the 2-category of all stacks under the localisation 2-functor of Pronk and Warren should be the 2-category of presentable stacks (I say should be, because I think I once checked this, and have promised to publish this result for a long time!). So you can combine all these ingredients to get that the 2-category of presentable stacks is equivalent to the anafunctor bicategory. It's a story I really need to write up properly one day, and it is to my shame that it hasn't happened yet...


