The standing objection is that on this account the cross is a demonstration and nothing more, and that a demonstration does not require a crucifixion. If the change happens entirely in the observer, a critic argues, then what is being observed could in principle have been anything sufficiently moving, and the actual death does no work that a lesser sign could not have done.
A second objection follows from the first. If nothing is accomplished outside the person who responds, then those who do not respond are in exactly the position they were in before, and the atonement turns out to be an occasion rather than an act. Critics call this subjectivism and hold that it makes salvation depend on the quality of a human reaction.
A third objection is historical and concerns whether the theory has been read fairly. Abelard's defenders argue that the exemplarist reading was fixed by his opponents and has been repeated ever since without being checked against the commentary, that he retained redemption language, and that the excursus on Romans is a short passage in a long work and was never offered as a complete system.
The defence, when it is made positively rather than as a correction of the record, runs like this. Every account of the atonement has to say something about how a past event reaches a present person, and the transactional models are weakest at exactly that point, since they explain what was done and leave the application to a separate doctrine. Moral influence starts where the others end. Its advocates hold that this is not a weakness but a different order of explanation, and that a theory which changes nobody has not explained the atonement either.
The atlas records that the objection is serious, that the reply is serious, and that no account of the atonement has ever been defined by a council, so neither side is arguing from a settled position.