Henry VII never met Catherine Howard.
Catherine was the wife of Henry VIII, not Henry VII.
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Henry VIII and Catherine Howard had no children. Jane Seymour was the mother of Henry's only legitimate son, Edward, who became King Edward VI.
She refused to bear him a son, he blamed her for this.
His eldest son's name was Arthur, he married to Katherine, died, and his brother Henry (It was common to name your son after yourself at this time), became King Henry VIII and married Arthur's widowed wife
Henry VIII wanted to marry Catherine of Aragon. After the death of Arthur Henry VII did not want Catherine or the money she brought to leave England so she was engaged to marry Henry. Henry VII however realized that Henry could contract a better marriage so there were talks that he would be engaged to one of Catherine's neices who was the daughter of Joanna of Castile and her husband Phillip the son of the Holy Roman Emperor.
Henry VIII was the son, and successor of Henry VII.