# Confusion in the definition of epimorphism of morphism of categories fibered in groupoids

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

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.