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][1], the answer would be yes -- but are there any references for this result in the literature? Thanks!


  [1]: https://mathoverflow.net/questions/16031/suppose-c-and-d-are-morita-equivalent-fusion-categories-can-you-say-anything-ab