How is an action potential transmitted across a cholinergic synapse?

Action potentials are depolarisations above the threshold level traveling along a neurone, communication often evolves the action potential being transmitted from one neurone to another, this is done via nerve synapse. 
The action potential will arrive in the presynaptic bulb, as an action potential has a positive potential difference this causes voltage gated Ca2+ ion channels to open. Ca2+ then diffuses into the pre-synaptic bulb by diffusion, causing vesicles containing the neurotransmitter, in this case acetylcholine (Ach) to move to the pre-synaptic membrane, acetylcholine is then released from the presynaptic bulb into the synaptic cleft by exocytosis.
The Ach then diffuses across the synaptic cleft and binds to receptor gated sodium ion channels this causes them to open and sodium diffuses across the post synaptic membrane into the post synaptic neurone creating an excitatory post synaptic potential, if this above the threshold level an action potential will be propagated along the post synaptic neurone. 
The enzyme acetylcholinesterase then degrades any leftover acetylcholine in the synaptic cleft and the products of this process then diffuse back into the presynaptic bulb and using ATP act as the start material for acetylcholine synthesis.