Informally we use the notion of a class. Given a formula $\phi (x, p_1, \ldots , p_ n)$, we call
a class. A class is easier to manipulate than the formula that defines it, but it is not strictly speaking a mathematical object. For example, if $R$ is a ring, then we may consider the class of all $R$-modules (since after all we may translate the sentence “$M$ is an $R$-module” into a formula in set theory, which then defines a class). A proper class is a class which is not a set.
In this way we may consider the category of $R$-modules, which is a “big” category—in other words, it has a proper class of objects. Similarly, we may consider the “big” category of schemes, the “big” category of rings, etc.
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