In general (Palanites are an exception due to a FAQ answer), each fighter lists what they can equip in their rules block, usually with some combination of "XYZ weapon types from this fighter's list" and/or "XYZ weapon types from the trading post". You have to check the fighter in question to find out what they can equip. This is why, as an example, Redemptionist Deacons from Cawdor can't use a Cawdor Heavy Crossbow. Here's their relevant ruleset:So trade is possible but equipping is only possible if both trading partners can have access to the same equipment?
Since the Cawdor Heavy Crossbow isn't on the fighter's equipment list and isn't in the Trading Post, the fighter can't equip it even if you have a spare one in your stash.A Redemptionist Deacon may purchase weapons and Wargear from the Redemptionist Deacon equipment list:
- During the course of a campaign, a Redemptionist Deacon may be given additional weapons and Wargear purchased from this list, from the Trading Post and from the Black Market.

