Step 1: Understanding the Concept
After a neurotransmitter has acted on the postsynaptic membrane, it must be removed quickly.
Step 2: Removal of the neurotransmitter
The enzyme cholinesterase in the synaptic cleft hydrolyses acetylcholine into choline and acetic acid. Choline is taken back for reuse.
Step 3: Why the other options fail
Adenylate cyclase makes cyclic AMP. Cholinergic fibres are nerve fibres, and catecholamines are themselves transmitters or hormones.
Final Answer:
The neurotransmitter is destroyed by cholinesterase.
\[ \boxed{\text{(B) }\text{cholinesterase}} \]