What an absolutely daft question.
Before flight, it is the pilots responsibility to check that the aircraft is properly registered, is airworthy and has been maintained properly.
The commander is required to carry out certain pre-flight actions as set out in Art 52. There is no generic requirement as stated.
a) Certificate of Airworthiness
is definitely a requirement, though there is no requirement specifically to check it (the commander can take it on trust, though would be committing an offence if there were no valid C of A).
b) Technical Log
is a nonsense. Only aircraft operated for public transport (including hire) or aerial work are required to even have a Technical Log. If one exists, it makes sense to check it to assist in making the assessment 'that the aircraft is in every way fit for the intended flight', but the commander is free to make an independent assessment.
c) Minimum Equipment List
is also not required for a private aircraft and most don't have them. That may be why this answer has been chosen. Art 19/Schedule 4 set out the minimum equipment required, and the commander is required to ensure that it is carried and fit for use.
d) Third Party Insurance Certificate
is not the responsibility of the commander, but of the aircraft operator.
Ironically, the only document that the commander
must check isn't listed: the certificate of maintenance review if one is required (again, not required for private aircraft).