Lighter wouldn't really hurt for backing, you just don't want much larger diameter monofilament than the braid equivalent, depending mostly how much mono you decide to use. In your case 20lb braid is roughly equivalent to 8 lb mono. So use 8, 10, 6, anything around there. If your filling a spool half mono, half braid, you obviously wouldn't want to use real heavy mono, like 17lb test or higher, and 20lb braid, as the braid is much thinner and will dig into the mono. If your just putting a real thin backing on the reel to ensure the braid dosent slip, I wouldn't worry too much about the differences in that case.
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