Given a manifold $M$ we have a geometric stack associated to it namely $\underline{M}$ whose objects are smooth maps to $M$. For the sake of consistency I am writing $BM$ for $\underline{M}$.

Given a Lie group $G$ we have a geometric stack associated to it namely $BG$ whose objects are principal $G$ bundles.

Given a Lie groupoid $\mathcal{G}$ we have a geometric stack associated to it namely $B\mathcal{G}$ whose objects are principal $\mathcal{G}$ bundles.

These are called as Yoneda embedddings (I do not have precise reference where it is called so, except for manifolds corollary $4.16$).

Given a smooth map $f:M\rightarrow N$, **if it is a submersion**, then, $M\times_NM$ is a manifold. We have $2$-fibre product $\underline{M}\times_{\underline{N}}\underline{N}$ and the stack $\underline{M\times_NM}$.

I am able to see that $\underline{M\times_NM}\cong \underline{M}\times_{\underline{N}}\underline{M}$. We have $B(M\times_NM)\cong BM\times_{BN}BN$.

David Roberts say here that same holds in case of Lie groups and Dimitri Pavlov say here that same holds for Lie groupoids i.e., we have following.

Given a morphism of Lie groups $\theta:G\rightarrow H$

**which is a surjective submersion**(submersion is to ensure $G\times_H G$ is a Lie group), then $$B(G\times_HG)\cong BG\times_{BH}BG.$$Given a morphism of Lie groupoids $f:\mathcal{G}\rightarrow \mathcal{H}$ such that the fibered product (in page no $5$, section $2.3$) $\mathcal{G}\times_{\mathcal{H}}\mathcal{G}$ is a Lie groupoid, then $$B(\mathcal{G}\times_{\mathcal{H}}\mathcal{G})\cong B\mathcal{G}\times_{B\mathcal{H}}B\mathcal{G}$$

Dmitri Pavlov said here that this has something to do with **Preservation of limits by the Yoneda embedding** and suggested this and this. But I am not familiar with $(\infty,1)$ categories. So, I am asking here (I am asking as a separate question).

How does one see that Yoneda embedding preserves limits in this setup? Please see my answer, I see the case in classical category theory.

Just an outline is also ok, just that it would be good if it is not mixed with $(\infty,1)$ categories.

presheaves. The 2-functor $LieGroupoids \to DiffStacks$ is not a full embedding, since not all maps between stacks come (even up to isomorphism) from functors between Lie groupoids. What you describe is the composite of $LieGroupoids \to Gpd^{Mfld^{op}}$, the 2-categorical Yoneda embedding, and stackification $Gpd^{Mfld^{op}} \to Stack(Mfld)$, and happens to factor through $DiffStacks \subset Stack(Mfld)$. $\endgroup$ – David Roberts Jan 22 at 23:25Given a morphism of Lie groups $\theta\colon G\to H$ which is a submersion<-- no, when $\theta$ is asurjectivesubmersion. When you say "it turns out", you should link to your source for this to aid other people who are learning, and so that people can check your assertion. :-) \\Given a morphism of Lie groupoids $f\colon \mathcal{G} \to \mathcal{H}$ that intersects transversally<-- this is meaningless, what does it mean for a single morphism to "intersect transversally"? Section 2.3 of the notes of Moerdijk to which you link never uses the word "transversal[ly]". $\endgroup$ – David Roberts Jan 22 at 23:35