  Given a not necessarily semisimple modular tensor category C, we use the
corresponding 3d TFT defined in [arXiv:1912.02063] to explicitly describe a
modular functor as a symmetric monoidal 2-functor from a 2-category of oriented
bordisms to a 2-category of finite linear categories. This recovers a result by
Lyubashenko [arXiv:hep-th/9405168] obtained via generators and relations.
Pulling back the modular functor for C to a 2-category of bordisms with
orientation reversing involution cancels the gluing anomaly, and further
pulling back to the original bordism category along a doubling functor leads to
the modular functor for the Drinfeld centre Z(C).
