Nylon or dacron webbing will work.
I've seen them fastened a couple different ways.
One end is usually an eyesplice and either fastened through a padeye or over a cleat luggage tag style, where the eye splice passes through the hole in the padeye or base of the cleat and then the rest of the line is passed through the eyesplice and then the eyesplice is snugged up around the padeye or cleat.
The other end can be an eyesplice that is brought through a cleat and then passed over the top of the cleat or it can be a plain end that is put around a cleat and cleat hitched and then has the bitter end seized to the standing part and the webbing brought up over the whole thing and sewn in place.
Another option is to attach the line using a privileged cowhitch and then lockstitch/seize the bitter end to the standing part of the line. This would work on either a cleat or padeye.
Privileged Cow Hitch:
For additional security, instead of just seizing and lock stitching the bitter end, you could pass it through the standing part ala a brummel splice and then seize/lock stitch it.
Start of brummel splice in 12 strand line:
Then snug the webbing up around the line as tightly as possible.
I hope this helps.
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