how to pull (HDMI) cable through existing conduit?

As long as you aren't using a unidirectional cable like a redmere then it doesn't matter what end you pull from. Just use electrical tape or some other strong tape to attach the ribbon to the cable. If you want to have another ribbon available add that at the same time you are pulling the cable i.e. you will be taping the cable and an extra ribbon to the existing ribbon and then pulling both through..

...or any lightweight string that you then attach to the ribbon at the other end and pull the ribbon back through.

It's not a loop, it's a pull string that's been secured at both ends, either by duct tape (I think I see it in the 2nd image), or by pinching it in a cut along the side of the conduit.

Protect the HDMI connector by wrapping it in paper and taping around it, secure the pull string to the cable behind the connector. Attach a new pull string to be pulled along (so you have something to pull the next cable that needs to go in). Have someone feed the HDMI cable into the conduit at one end while another person slowly pulls the string at the other end.

You can get nylon pull string at Home Depot / Lowe's, or a spare run of cat5e can be used, too. Although if you're pulling anything, you should pull at least 1 or 2 Cat5e cables along with your HDMI, which should help cover you for the future. Pulling additional cables later will always be harder than pulling a bunch at once.

Jeff

You are going to want to observe the maximum pulling tension for the cable. Pulling harder can affect the geometry of the cable, or damage it, affecting its performance. Pulling an HDMI cable is not the same as pulling electrical conductors. It may be hard to find out what the maximum pulling tension for your cable is. For BJC Series 1, it is 60 pounds. For smaller wire, it is going to be less.

I didn't take any chances and used the pull strings to pull an electrical fish tape through the conduit, then used the fish tape to pull all my wires. I secured the more sensitive HDMI cables to the tape by wrapping the ends of the tape/cable with electrical tape so I had 12" of bearing on the cable. Probably overkill, but electrical tape is cheaper and faster than replacing a bum cable. Like others suggested, just add another pull string to the new cable being pulled so you have one later to use.

Use gravity, pull down. Use 2 people, 1 at each end. Pull all cables together, if more than 1. Cable lube is available, check HD/Lowes.