Mechanism for synthesising azetidine
There is not much to say other than: your version is more or less correct. It is debateable whether the ester hydrolysis happens before or after the cleavage of azetidine (I would suspect before due to the lower overall charge). Remember that you always have to consider aquaeous work-up at the end of a reaction, which will give you the three protons you need to regenerate acrylic acid, ethanol and azetidine.

You also have another question written on your paper after the halogenation of the terminal alcohol, namely:


  What is the point of this step?


It’s not really a point. It’s just that the reaction conditions contain $\ce{HCl}$ which will protonate any strong enough base around and the secondary amine is a strong enough base. It’s more of a consequence of the reaction conditions than an actual pointful step.