Does the Synchronize ability control a caught Pokémon's nature and if so, how does it work?
Effect
If the opponent causes a burn, paralysis, or poisoning of a Pokémon with Synchronize, the opponent receives the status condition too. Self-inflicted status conditions (for example through the use of an item) are not passed on.
Changes
In Generations 3-4, bad poison is only passed on as regular poison.
Outside of battle
If a Pokémon with Synchronize is the leading party Pokémon, there is a 50% chance of each wild Pokémon encountered having the same nature.
Source