Investigating the interplay between initial hard processes and final-state effects measuring prompt and non-prompt $J/\psi$

Charmonium production in high-energy collisions can be split into a prompt and a non-prompt component. Both components can be distinguished experimentally by studying displaced topology, and represent valuable tools to investigate the properties of the strongly interacting medium produced in ultra-relativistic heavy-ion collisions. In particular, the study of non-prompt charmonium production can give access to the beauty-hadron production cross section and can be used to investigate the in-medium energy loss of beauty quarks. In these proceedings, the recent measurement of prompt and non-prompt $J/\psi$ carried out by the ALICE Collaboration in PbPb collisions at midrapidity ($|y|$ < 0.8) at $\sqrt{s_{\rm NN}}=5.02$ TeV, will be presented. Moreover, thanks to the installation of the new muon forward tracker (MFT), the prompt/non-prompt charmonium separation will be possible in LHC Run 3 also at forward rapidity (2.5 < $y$ < 4).