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Related to fasten up: fasten down
To secure or close something, as by using buttons, zippers, etc. A noun or pronoun can be used between "fasten" and "up." It's freezing today, so be sure to fasten up your coat.
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.