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    $\begingroup$ The question is vague. What is "the issue"? The universe axiom is sufficient for most purposes – I have yet to encounter a situation where it is truly and unavoidably necessary to ask for a category of all sets or whatever. $\endgroup$ Commented Feb 19, 2014 at 22:13
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    $\begingroup$ Well, the category of all sets (or else) is sometimes necessary. I do have found such things in well established mathematical theories. Nevertheless, what's the problem of working with proper classes? $\endgroup$ Commented Feb 19, 2014 at 22:27
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    $\begingroup$ @FernandoMuro Working with classes is subtle. For instance, it is not possible to quantify over classes, let alone form collections of classes. In particular, there is no such thing as the category of all functors $\mathbf{Set} \to \mathbf{Set}$ if $\mathbf{Set}$ is genuinely the category of all sets. For the working mathematician, it is better to use the universe axiom than to worry about the finer details of logic. $\endgroup$ Commented Feb 20, 2014 at 9:32
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    $\begingroup$ Dear @Fernando, may I ask you to provide an example illustrating your claim? $\endgroup$ Commented Feb 21, 2014 at 19:25
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    $\begingroup$ @arsmath Yes, I am perfectly aware. I did write a whole article on the subject, after all. In my mind, the purpose of the universe axiom is to allow us to treat sets, classes, collections of classes, collections of collections of classes, etc. all on the same basis. $\endgroup$ Commented Mar 23, 2014 at 16:07