In the parallel universe that Kochanski came from, Lister was very similar to his primary universe counterpart - to begin with.
The divergence between the two versions of Lister occured when Kochanski had broken off their relationship. In his universe, she felt guilty about dumping him in a letter and went to see him in the locker room area on Red Dwarf after he had returned from shore leave on Mimas. She apologised for the note and attempted to make amends, until she noticed the small kitten that Lister had smuggled aboard.
She insisted that the cat should be destroyed, but couldn't bring herself to disintegrate a kitten. She kept the cat as her own, meaning it was she who was caught and sent to stasis. Lister was killed in the Cadmium II leak and Kochanski was revived 3 million years later. Holly brought Lister back as a hologram to keep her sane.
Being a soft light hologram for several years made Lister a much more sensitive and caring person. He and Kochanski revived their romance after time and he was prepared to have long talks about sensitive subjects like emotions and relationships. He also became very cultured and intelligent, playing games like "The Magic Flute" and knowing how to operate complex systems.
He wore a sophisticated black suit and had an "H" on his forehead, but was otherwise identical in appearance to his primary counterpart. (RD: Ouroboros)
When a spatial rift created a dimensional linkway between their two realities, the two Listers met when the crews went into non-space. They were soon separated by GELF weapons fire that severed the connection, with Kochanski on the wrong side. She told her new shipmates all about her boyfriend, making Lister extremely jealous.
Kryten described this version of Lister as "every woman's dream guy". He enjoyed shopping for shoes and had even given up his reliance on spicy food once he had obtained a hard light drive.
When the linkway was reestablished, the hologram Lister brought several supply crates with him, bemoaning how much curry was included. He was distraught when he thought that Kochanski had died in non-space. Once Kochanski had moved to the primary Starbug, she looked for ways to get back to her universe. (RD: Ouroboros)
The human Lister described his counterpart as "The Anti-Lister", owing to his sensitivity and lack of interest in spicy food. He especially hated the way Kochanski was always saying what her Lister would have done in any given situation. She would often tell her new shipmates about one of his sayings or another aspect of his caring nature. Lister mockingly called his hologram counterpart "the hologrammatic hunk" (RD: Blue)