Then he'd have read Horace's probably barely passing psych reports. He'll be very watchful of him, but beyond that... There won't be any negative interaction unless Horace starts it. And... It seems that he will with that pistol.
Though I do have to voice a little concern about his '6th sense' it seems pretty meta-gamey to me, but... It's not really doing anything for right now and with zeds all over the place, he'd probably have his gun out anyway.
And as far as their past relationship goes... If there were suspicions about Horace killing his own men, Carter would have read interrogation reports and whatnot, and any other nasty activity would have been the one to dole out punishments. However he's very civil. Would state the punishment, ask the soldier if they understood why it had to be done, and if they planned on doing it again. And then why they answered what they did, whether it was yes or not. Along with asking why they did what they did.