Failing to understand set-theoretic issue in Borceux The following is an excerpt of "Handbook of Categorical Algebra 2" by Borceaux, chapter 8 on fibered categories.

Working with set-indexed families as in the previous example very often leads to constructions or arguments which look innocent... but are formally incorrect! For example given an $I$-indexed family of morphisms $(f_i : C_i \to D_i)_{i \in I}$ in $\mathsf{Set}(\mathscr{C})$, few people really worry about considering the "set"
$$I_0 = \{ i \in I \mid f_i \text{ is a monomorphism} \}.$$
But the sentence "$f_i$ is a monomorphism" means
$$ \forall C \in \mathscr{C} \ \forall u,v \in \mathscr{C}(C, C_i) \quad f_i \circ u = f_i \circ v \implies u = v$$
... and no axiom of set theory will ever imply that $I_0$ is a set, since the formula contains a quantifier acting on a variable $C$ which runs through something (namely, $\mathscr{C}$) which is not a set!

I am terrible at foundations, and I absolutely do not understand the arguement he is making here.
So, the $C_i, D_i$ are objects in $\mathcal{C}$, and $I$ is a set (that's by definition).
In particular, $I_0$ as defined there is a "subclass" of $I$, and hence a set, no?
Or is this some sort of decidability-issue? That is, membership in $I_0$ is not decidable in our (smallish) universe, and therefore we cannot properly build $I_0$?
I'd also be happy to get pointers at some resources where such issues are properly discussed, and not only marginally :(
 A: I think Borceux's argument is wrong. Recall Comprehension scheme 1.1.8 from Volume I:
"If $\phi(x_1,...,x_n)$ is a formula where quantification just occurs on set variables, there exists a class $A$ such that $(x_1,\dots,x_n)\in A$ if and only if $\phi(x_1,\dots,x_n)$." Quantification over sets in the context of Bernays-Godel set theory means bounded quantification, e.g. having "$\forall x\in C$" instead of "$\forall x$" (this is because $x\in C$ implies $x$ is a set). Borceux seems to erroneosuly think that $C$ itself in "$\forall x\in C$" should be a set instead of a class for the comprehension scheme to apply.
The confusiong perhaps stems from not formulating the more general comprehension scheme with parameters: given $\phi(x_1,\dots,x_n,Y_1,\dots,Y_m)$ with bounded quantifiers, then $\forall Y_1,\dots,Y_m\exists A((x_1,\dots,x_n)\in A\iff\phi(x_1,\dots,x_n,Y_1,\dots,Y_m)$. Here $x_1,\dots,x_n$ should be set-variables, i.e. we are assuming $\phi(x_1,\dots,x_n,Y_1,\dots,Y_m\implies\exists Z_i(x_i\in Z_i)$.
Now, clearly the sentence that "$f_i$ is a monomorphism", i.e. "$\forall C \in \mathscr{C} \ \forall u,v \in \mathscr{C}(C, C_i) \quad f_i \circ u = f_i \circ v \implies u = v$", involves only bounded quantification. So by 1.1.8 there is a class of monomorphisms in any category. Presuming that $Set(\mathscr C)$ is a category (which it is because set-indexed families of elements of a class form a class), $I_0=\{i\in I:f_i$ is a monomorphism$\}$ is then a subclass of $I$ (since "$f_i$ is a monomorphism" is a class) and is therefore a set.
