Relationship between fusion category and its Drinfel'd center Is it true that given a fusion category $\mathcal{C}$ and its Drinfel'd center $Z(\mathcal{C})$, there is a fully faithful functor $F:\mathcal{C}\hookrightarrow Z(\mathcal{C})$? I.e. can $\mathcal{C}$ can be identified with a full monoidal subcategory of $Z(\mathcal{C})$?
Judging by the mention of a 'restriction functor' $Z(\mathcal{C})\to\mathcal{C}$ here, the answer would be yes -- but are there any references for this result in the literature? Thanks!
 A: I assume we are under the assumption the category is braided.
This is mentioned in
Drinfeld, Vladimir; Gelaki, Shlomo; Nikshych, Dmitri; Ostrik, Victor, On braided fusion categories. I, Sel. Math., New Ser. 16, No. 1, 1-119 (2010). ZBL1201.18005.
right after Proposition 2.36 , and in EGNO Proposition 8.6.1 with very similar wording. Both without proofs and without attribution.
The proof seems straightforward. The assignment is $X\to (X,R_{\bullet,X})$ where $R_{\bullet,X}$ is the braiding $Y\otimes X\to X\otimes Y$.
Morphisms in the Drinfeld center between objects $(X,\gamma^{X})$ and $(Y,\gamma^{Y})$ are given by morphisms $f:X\to Y$ such that $\gamma^{Y}_{M}\circ Id_{M}\otimes f = f\otimes Id_{M} \circ \gamma^{X}_{M}$ for an object $M$. But in our case $R_{\bullet, X}$ is the braiding of the category so any morphism $X\to Y$ works.
Edit:
In the nonbraided case there is a right adjoint to the forgetful functor. This is again already mentioned in ENO (5.8) and EGNO (9.2, for example) without attribution and for what I can see, also no proof that this is a right adjoint ( but follows from the adjunction formula for the internal Hom ).
You can however prove that the induction functor $I:C\to Z(C)$ is given by $I(X)\cong \underline{Hom}_{Z(C)}(1,X)$ in EGNO Proposition 8.12.1.
But I don't think you get that this functor is fully faithful, in fact the relevant result is that it induces an equivalence of $C$ with the category of A-modules in $Z(C)$ where $A=I(1)$ (EGNO Proposition 8.12.2 (ii)). When the category is braided you get that this category of modules is equivalent to the center, but I don't think this is true in general. I admit I cannot think of a counterexample, but hope this is of any help.
