Suppose we work on the Grothendieck site $\mathcal{C}$ of all schemes in the fpqc topology. If it helps it is also fine with me to work only over affine schemes.
Let us denote the category of stacks over $\mathcal{C}$ by $Stacks(\mathcal{C})$. This is naturally a full subcategory of the category $Pre\_{grpds}(\mathcal{C})$ of presheaves in groupoids over $\mathcal{C}$. This is equivalent to formulating stacks as being categories fibered in groupoids, fulfilling effective descent.


Now the inclusion $Stacks(\mathcal{C}) \to Pre\_{grpds}(\mathcal{C})$ has a left adjoint called the stackification functor.


It is a classical fact that the sheafification functor on presheaves of sets commutes with finite limits. Is this also true for the mentioned stackification functor?


Precisely, I want to know if and if so, why, the stackification functor commutes with 2-categorical pullbacks, here is the precise setting.


Suppose we are given Hopf-algebroids $P\_1,P\_2,P$ and view them as presheaves in groupoids.
Suppose we are given two morphisms $P \to P\_i$ and we construct the 2-pullback of functors $$Q = pullback(P\_1 \to P \leftarrow P\_2).$$


In my situation I can show that Q is equivalent to an affine scheme, and I want to conclude that the pullback of the stackified diagram is also equivalent to an affine scheme.
If I knew that stackification commuted with finite limits, this would be ok, since the (co)unit (I always mix them up) of the adjunction is an equivalence, i.e., if $X$ is a stack, and $st(X)$ is its stackification, then the natural map $X\to st(X)$ adjoint to the identity of $st(X)$ is an equivalence of stacks.


Any help and comments are appreciated, also if you could give references concerning these questions, that would be great.



stackification commutes with finite limits?
The answer is yes, at least for $2$-fiber products. And fortunately there is an excellent reference online: [Tag04Y1](https://stacks.math.columbia.edu/tag/04Y1) in the Stacks Project. I quote:



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> Lemma 8.4. Let $C$ be a site. Let $f : X \to Y$ and $g : Z \to Y$ be morphisms of ﬁbred categories over $C$. In this case the stackiﬁcation of the 2-ﬁbre product is the 2-ﬁbre product of the stackiﬁcations.
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