I'm in a Tree Magic, the bridge is a wide (4cm) webbing strap, adjustable for length. It's double layer, not stitched. Nylon I guess.
In my avatar that's a Butterfly with two bridges, the original thick webbing bridge and the replaceable B'fly2 thinner webbing bridge with a cover (always thought I should get rid of the cover...). Both are triple layer stitched, the original is wide and thick, the newer one thinner and narrower, and comes with a ring so it got promoted to primary use with the HC and VT