Question about the proof of the completeness of the category of rooted cluster algebras.

Question

I have been reading the proof of Propostion 5.4 in On the category of rooted cluster algebras by Assem, Dupont and Schiffler in which the authors state that the category of rooted cluster algebra is not complete. They prove it by the means of a counter example. I am struggling to understand their proof. I am very confused with the last step, to be specific. They somehow deduce there that a certain element of a certain ring is mapped to two different elements, which is a contradiction. But by the assumption at the start, we know what this element is actually mapped to but they somehow seem to disregard that fact or maybe, which is probably much more likely, I am just not seeing something. I know this is very generic but maybe there is someone out there who is familiar with the topic and willing to shed some light on this.