fasten someone or something (on)to someone or something
to attach people or things together. He fastened himself onto the mast of the boat with a length of rope. I fastened a note onto Jimmy, so his kindergarten teacher would read it and remind him to wear his gloves home.
fasten something down (to something)
to attach something down to something else. Fasten this board down to the top of the workbench. Please fasten down the board.
fasten something up
to close something up, using buttons, a zipper, snaps, hooks, a clasp, or other things meant to hold something closed. Please fasten this up for me. I can't reach the zipper. Please fasten up my buttons in back.
fasten (up)on someone or something
1. Lit. to take firm hold of someone or something. She fastened upon me and would not let me go until she finished speaking. I don't like people who fasten on me and ask a lot of questions.
2. Fig. to fix one's attention on someone or something. He fastened upon the picture for a brief moment and then turned away. The baby fastened on the television screen and watched it for many minutes.