This paper examines proof constructions in group work in the field of linear algebra teaching at the university level. Studies have shown that students at tertiary level have difficulties in understanding different kinds of quantifiers, which are fundamental in linear algebra proof constructions. This study investigates how two student groups, with a tutor involved in one of the groups, construed meaning in the context of proving unique existence of the adjoint endomorphism. The students and the tutor used certain words and phrases in the context of mathematical uniqueness differently. The study analyses from an interactionist standpoint how these ambiguities emerged. The results indicate that due to different background understandings of mathematical uniqueness students attributed different meanings to certain words and expressions, which prevented the students from negotiating a consensus during the proving process.