Covering morphisms of topological internal groupoids

Let $X$ be a topological group with operations whose underlying space has a universal cover. Then the fundamental groupoid $\pi X$ becomes a topological internal groupoid, i.e., an internal groupoid in the category of topological groups. In this paper, we prove that the slice category $\text{Cov}_{sTC}/X$ of covering morphisms $p:\tilde{X}\rightarrow X$ of topological groups with operations in which $\tilde{X}$ has also a universal cover and the category $\text{Cov}_{Gpdinsert ignore into journalissuearticles values(TC);}/\pi X$ of covering morphisms $q:\tilde{G}\rightarrow \pi X $ of topological internal groupoids based on $\pi X$ are equivalent. We also prove that for a topological internal groupoid $G$, the category $\text{Cov}_{Gpdinsert ignore into journalissuearticles values(TC);}/G$ of covering morphisms of topological internal groupoids based on $G$ and the category $\text{ACT}_{Gpdinsert ignore into journalissuearticles values(TC);}/G$ of topological internal groupoid actions of $G$ on topological groups with operations are equivalent.