Well it is certainly a Cotoneaster but I'm not certain it is horizontals even having turned the lap top upside down to look at the photo. Doesn't really matter as cuttings from Cotoneasters are taken from semi-ripe wood, I'd have thought it was a little late now but here is the link to the RHS instructions:

http://apps.rhs.org.uk/advicesearch/Profile.aspx?pid=404

You'd probably be better off taking a load of berries off and growing it from seed....make sure the seeds gets a good cold spell. Not horizontalis as per above but it is lovely and the blackbirds will love those berries, so get the seeds before they do !!!!

Hi, and welcome to G o y, I agree with the others, definitely not C horizontalis, that grows, as the name implies, horizontally accross the ground, and is not very tall, only to about 3 ft max x a spread of about 5 ft, but as said, you can collect the seed, put them in a tin with a lid, and bury them until late winter/ early spring, them sow them in a seed tray and keep them in a cold frame, pot them up when large enough to handle, and grow on until the following autumn and plant out, Derek.

It looks like a C. dammeri relative, but the propagation should be the same as C. horizontalis.

Have a look on the ground where the branches have reached, there may well be self rooted plants which can be severed from the parent plant.