Linking first and second level composite primary keys
You need 2 columns in the foreign key constraint, to refer to a table with  2 columns in the primary key:

-- Second higher level combining 
CREATE TABLE full_b_c(
    b_ID INT NOT NULL,
    c_ID INT NOT NULL,
    a_ID INT NOT NULL references a ON UPDATE CASCADE,
    PRIMARY KEY (b_ID, c_ID, a_ID),
    FOREIGN KEY (b_ID, c_ID) 
      references b_c (b_ID, c_ID) ON UPDATE CASCADE
);