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# Fibered product of stacks comes from a Lie groupoid

Suppose $$\mathcal{G},\mathcal{H}$$ are Lie groupoids and $$F:B\mathcal{G}\rightarrow B\mathcal{H}$$ be a morphism of stacks.

We can talk about the fibered product $$B\mathcal{G}\times_{B\mathcal{H}}B\mathcal{G}$$.

Is it true that this stack is also of the form $$B\mathcal{K}$$ for some Lie groupoid $$\mathcal{K}$$?

We can ask the same question in the set up of Lie groups. I guess answer would be same in both cases.

Suppose $$G,H$$ be Lie groups and $$F:BG\rightarrow BH$$ be a map.

We can talk about the fibered product (pullback) $$BG\times_{BH}BG$$.

Is it true that this space is also of the form $$BK$$ for some Lie group $$K$$?

Suppose we have a map of Lie groups $$G\rightarrow H$$ inducing the map $$F:BG\rightarrow BH$$ then, I am guessing that $$G\times_HG$$ (if it is a Lie group) is the group $$K$$ I am asking above.