Thank you for the anwer. Could you explain why there is a degree 1 line bundle which is not effective.

Thank you for your answer! It was very convincing from a philosophical point of view.

@abx What you are saying is completely correct. We can interpret all claims in the proper sense. However, I am afraid that the above assertion about the smallness of the pair may produces errors in various places. So I would like to know if I am wrong in the above discussion.

@LSpice My source is exactly the part pointed out by abx.

@abx If that were the case, we would have to treat a category as an ordered pair of a set of morohisms, compositions, and so on. In that case, we end up with the same problem.

@Zhen Lin In page 2 of SGA 4 (see abx's link above), he says 'On dit qu’un ensemble est U-petit (ou, quand aucune confusion n’en résulte, petit) s’il est isomorphe à un élément de U.'