This is a follow up to this question.
In view of Luca Bressan's answer it seems that Definition 2.1 in the first Exposé of SGA4 doesn't really make sense as stated. Luca Bressan found a fix which, I think, is probably the one that Grothendieck and Verdier would have given if the problem had been pointed out to them. There is, however, a putative fix which is much simpler, and indeed so simple that I suspect there is a catch. But I haven't been able to find this catch. So, I will describe the putative fix and ask the reader to tell me what's wrong with it.
First of all, to avoid any misunderstanding, let me spell out an important assumption which is implicit in the Exposé: categories have, by definition, disjoint hom-sets.
For any universe $\mathcal U$ and $\mathcal V$ such that $\mathcal{U\subset V}$ write $i_{\mathcal{VU}}$ for the inclusion of $\mathcal U\text{-Set}$ into $\mathcal V\text{-Set}$.
The basic principle can be summarized as follows. In order to develop category theory we have to make lots of choices of universe, but we want of course our end results to be independent of these choices.
The following construction turns out to be a crucial tool. To a category $\mathcal C$ and a universe $\mathcal U$ we attach the category $\widehat{\mathcal C}_{\mathcal U}$ of functors $F:\mathcal C^{\text{op}}\to\mathcal U\text{-Set}$, such functors being called $\mathcal U$-Set valued presheaves over $\mathcal C$, or just presheaves.
Let $F\in\widehat{\mathcal C}_{\mathcal U}$ and $G\in\widehat{\mathcal C}_{\mathcal V}$. We consider $F$ and $G$ to be "essentially equal" if $F(X)=G(X)$ for all $X$ in $\mathcal C$, or, equivalently, if there is a universe $\mathcal W$ such that $\mathcal W\supset\mathcal U,\mathcal V$ and $i_{\mathcal{WU}}\circ F=i_{\mathcal{WV}}\circ G$. In some sense we are interested in presheaves mainly up to "essential equality". This means that there is no harm in replacing the universe we are working with by a larger one.
In the beginning of the first Exposé of SGA4 the authors attach
$\bullet$ a presheaf $h_{\mathcal U}(X)$ to an object $X$ of a category $\mathcal C$ and
$\bullet$ a presheaf $\varprojlim^{\mathcal U}G$ to a functor $G:I\to\mathcal C$
by choosing a certain universe $\mathcal U$, the constraint being of course that these presheaves should be well defined up to "essential equality".
Grothendieck and Verdier choose a universe $\mathcal U$ which has the property
$(P(\mathcal U))$: for all objects $X$ and $Y$ of $\mathcal C$, the set $\text{Hom}_{\mathcal C}(X,Y)$ is equipotent to some element of $\mathcal U$.
We suggest to consider the stronger property
$(Q(\mathcal U))$: $\mathcal C\in\mathcal U$.
Note that $(P(\mathcal U))$ and $\mathcal U\subset\mathcal V$ imply $(P(\mathcal V))$, and similarly for $Q$.
My suggestion is to replace property $P$ with property $Q$.
If we do this, we don't need Axiom $(\mathcal U\ B)$, which is stated in the language of Bourbaki's set theory, and whose translation in the ZFC language is not obvious to me. Moreover, in "Construction-définition 1.3", we can define $h_{\mathcal U}(x)(y)$ as being $\text{Hom}_{\mathcal C}(y,x)$ is all cases. Finally, the definition of the projective limit of a functor $G:I\to\mathcal C$ given in Definition 2.1 of the Exposé becomes correct, even without Luca Bressan's fix. (We assume $I\in\mathcal U$ instead of "$I$ is $\mathcal U$-small"; any $\mathcal U$-small category being isomorphic to some category $\in\mathcal U$, we don't really lose generality. Note that the fact that $\varprojlim^{\mathcal U}G$ well defined up to "essential equality" is not stated explicitly by Grothendieck and Verdier.)
What's wrong with this approach?