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I am reading Differentiable Stacks and Gerbes by Kai Behrend and Ping Xu.

According to this paper, a morphism of groupoid fibrations (Categories fibered in groupoids over the category of manifolds) $F:\mathcal{D}\rightarrow \mathcal{C}$ is an epimorphism if for every map of stacks $\underline{U}\rightarrow \mathcal{C}$, where $U$ is a manifold, there exists a surjective submersion $V\rightarrow U$ and a $2$-commutative diagram enter image description here

Equivalently, $V$ may be replaced by an open cover of $U$, in this statement.

I am reading Principal actions of stacky Lie groupoids by Henrique Bursztyn, Francesco Noseda, Chenchang Zhu.

According to this paper, a morphism of categories fibered in groupoids over the category of manifolds $F:\mathcal{D}\rightarrow \mathcal{C}$ is an epimorphism if given a manifold $U$, there is a cover $\{U_\alpha\rightarrow U\}$ such that given $y\in \mathcal{C}(U)$ there exists for each $\alpha$ an element $a_\alpha\in \mathcal{D}(U_\alpha)$ with isomorphisms $F(a_\alpha)\rightarrow y|_{U_\alpha}$.

I am just trying to see if these two notions are same or not. Any coments are welcome.

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