Let $G$ be a finite group. Barwick et al define a $G-\infty$-category to be a fibration over the orbit category $O_G$ of transitive $G$-sets. But in the non-$\infty$-land, the natural guess at where I should work to do $G$-equivariant homotopy theory is a category enriched in the category $Top_G$ of $G$-spaces. In the $\infty$ world, this setting can be generalized directly using Gepner and Haugseng's notion of enriched $\infty$-category.

Question: Is there a comparison functor between $\infty$-categories fibered over $O_G$ and $\infty$-categories enriched in $Top_G$? Is this an equivalence, perhaps after passing to certain subcategories?

Let $G$ be a finite group. Barwick et al define a $G-\infty$-category to be a fibration over the orbit category $O_G$ of transitive $G$-sets. But in the non-$\infty$-land, the natural guess at where I should work to do $G$-equivariant homotopy theory is a category enriched in the category $Top_G$ of $G$-spaces. In the $\infty$ world, this setting can be generalized directly using Gepner and Haugseng's notion of enriched $\infty$-category.

Question: Is there a comparison functor between $\infty$-categories fibered over $O_G$ and $\infty$-categories enriched in $Top_G$? Is this an equivalence, perhaps after passing to certain subcategories?

EDIT: Maybe this is what Marc is driving at in the comments, but think of it this way. A category fibered over $O_G$ is a functor $O_G^{op} \to Cat$, which is a functor $O_G^{op} \times \Delta^{op} \to Top$ satisfying some conditions. A category internal to $Top_G = Fun(O_G^{op}, Top)$ is a simplicial object in $Top_G$, i.e. a functor $\Delta^{op}\times O_G^{op} \to Top$, satisfying some conditions. This leads me to post a

Revised Question: Are categories fibered over $O_G$ the same thing as categories internal to $Top_G$? Which ones correspond to enriched categories?