The correct answer is C) to the owner for vessel administration and to the charterer for cargo operations and schedule. In a time charter party, the charterer has temporary control and use of the vessel, while the owner retains ownership and responsibility for the vessel's administration, such as manning, maintenance, and compliance with regulations. The charterer is responsible for the cargo operations and scheduling the vessel's voyages during the charter period. The other options are incorrect because: A) the responsibility is not solely to the cargo shippers and consignees, B) the responsibility is not solely to the owner as in normal conditions, and D) the responsibility is not solely to the charterer for all matters pertaining to cargo and ship administration.