How does one solder a capacitor ?

Capacitor? With leads that short?

Hot solder on the tip of the iron, coat the wire, touch hot wire to lead, done.

Be quick and heat everything but the cap. They aren't exceptionally difficult to solder. Just don't heat it excessively.

Did you pull this off something used?

Typically when sold new, they have much longer leads, which makes soldering easier, and then you trim off the extra lead length you don't need.

With the leads this short, it would be very challenging even for an expert to solder this without heat damaging the capacitor.

Idk where youre from but in my native language its also basically "condensator" but in english theyre usually called capacitors. In some old compounds like condenser microphone you see the old form.

Just be careful of the polarity, and try to move quickly since the legs are so short

There's some YouTube videos on how to solder... For example this random one looks like the capacitor has been harvested and has very short leads: https://youtube.com/watch?v=iDtXq21bKYg

Basically: Get the polarity right, or it'll blow. Touch both the lead and the pad with your soldering iron, and after a second, add the tin solder, wait another like two seconds for it to melt and connect everything and you're done. Then check your work and see you didn't inadvertently short-circuit anything else. Shouldn't be too hard for a capacitor as the pads are quite far apart. And don't burn the capacitor by heating it up for way too long.

Pro-tip: Once you test the device, put the casing on to avoid touching mains voltage. Will also contain something like an exploding capacitor so you don't get it into your eyes. If it works you can tighten all the screws.

oh god