Each block's header contains a hash that is the root of a Merkle tree [0] of the transactions in the block. The Merkle-root hash effectively summarizes all of the block's transactions, which allows the overall block hash to be a hash just of the constituents of the header.

Thus if you have the header, you do indeed have everything you need to produce a hash and verify that it matches the hash referenced in the succeeding block. You do not need the information describing individual transactions.

0. https://en.m.wikipedia.org/wiki/Merkle_tree