A topic from AEG's L5R CCG Rules forum, copied for Oracle of the Void from a copy of the forum saved in October 2015, so pages it didn't save are missing. Players' names are left out; the rules officials are named. Rulings are as they stood on these dates; later rulings, the rulebooks and the Rules Archive may differ.
A player, 29 May 2008
Kakita Kaisei Crane Clan Samurai • Courtier • Duelist • Magistrate Reaction: After Kaisei issues a challenge: Target a Personality controlled by the challenged player. Refusing the challenge has the additional cost of Dishonoring the target. (If it already has the cost of Dishonoring that target, the challenge may not be refused.)
Impromptu Duel Iaijutsu Limited: Target one of your unbowed Samurai or Duelist Personalities: He challenges another player’s target Personality. The other plaer may refuse; if he does, dishonor his Personality and gain 4 Honor. Destroy the duel’s loser. The winner gains 2 Honor.
How does Kakita Kaisei work with Impromptu Duel?
It looks like dishonoring is an effect of refusing but I can't seem to find any duels that have dishonoring as a cost, what card(s) is the "(If it already has the cost of Dishonoring that target, the challenge may not be refused.)" text refering to?
Brook Cunningham (rules official), 29 May 2008
Kakita Kaisei Crane Clan Samurai • Courtier • Duelist • Magistrate Reaction: After Kaisei issues a challenge: Target a Personality controlled by the challenged player. Refusing the challenge has the additional cost of Dishonoring the target. (If it already has the cost of Dishonoring that target, the challenge may not be refused.)
Impromptu Duel Iaijutsu Limited: Target one of your unbowed Samurai or Duelist Personalities: He challenges another player’s target Personality. The other plaer may refuse; if he does, dishonor his Personality and gain 4 Honor. Destroy the duel’s loser. The winner gains 2 Honor.
How does Kakita Kaisei work with Impromptu Duel?
It looks like dishonoring is an effect of refusing...
I assume you're referring to Impromptu Duel in your last sentence quoted above, and that's correct. If you use Kaisei's Reaction to him issuing Impromptu Duel's challenge, an honorable Personality could pay the cost to refuse. Impromptu would then try to dishonor him as an effect for refusing, which would just fizzle if he is already dishonorable from paying the cost to refuse that Kaisei added.
...but I can't seem to find any duels that have dishonoring as a cost, what card(s) is the "(If it already has the cost of Dishonoring that target, the challenge may not be refused.)" text refering to?
Kaneka's Conflict, from the previous set.