You can certainly make the argument but you would have to prove prejudice. The main problem is that unless new facts are alleged, other than the assignment, prejudice is usually difficult to prove. And, generally, any prejudice you could allege would be ameliorated if the court simply continued the trial date.
IMO, now is the time for you to at least have counsel review the initial and proposed amended complaint to see if you have better arguments against the proposed amendment and any defenses to the newly proposed causes of action. If new facts are alleged perhaps the statute of limitations would bar the newly asserted causes of action. Or, perhaps a cause of action is not assignable.
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Generally a party can amend a complaint at any time, even during trial.
You may have defenses to the new claims, so you should at least take everything to a local attorney for a consultation.
In California, the courts are very liberal in granting leave to amend a complaint. If you have already been involved in the litigation as a cross-defendant, it is probably going to be very difficult to argue prejudice to you. Even if you got added as a new defendant in the main case, at most you can ask for is a trial continuance based upon the new claims.
Frank W. Chen has been licensed to practice law in California since 1988. The information presented here is general in nature and is not intended, nor should be construed, as legal advice for a particular case. This Avvo.com posting does not create any attorney-client relationship with the author. For specific advice about your particular situation, please consult with your own attorney.