I'm trying to understand Proposition 2.9 of this paper on weakly group theoretical fusion categories.

First of all I have a problem with understanding the settings for de-equivariantization process. It is written on page 5 of the paper that one needs $\mathcal{E}=\mathrm{Rep}(G)\subset Z(\mathcal{C})$ such that $\mathcal{E}$ embeds into $\mathcal{C}$ via the forgetful functor $Z(\mathcal{C})\rightarrow \mathcal{C}$.

**Question 1.**
I would like to know what "embeds" means exactly.

Rephrasing when a functor is an embedding functor is? Is it enough to be injective on objects and to send simple objects into simple objects. Obviously this is necessary.

**Question 2.** In a concrete example where $\mathcal{C}=\mathrm{Rep}(A)$ for a Hopf algebra $A$ what would be the conditions to be imposed for the corresponding composition to be an embedding.

Is the corresponding category $\mathcal{C}_G$ the category of representations of a Hopf algebra, i.e does it posses a fiber functor?

**Question 3.** The third question I have is regarding the de-equivariantization $\mathcal{E}'_G $ from the second part of the proof proposition.

I understood the construction of $\mathcal{E}$ and clearly $\mathcal{E}\subset \mathcal{E}'$ since $\mathcal{E}$ is symmetric. The question I have is why after composing to the restriction functor $Z(\mathcal{E}') \rightarrow \mathcal{E}'$ one still has an inclusion.