Let $C$ be a locally small category, so the objects form a class (in the sense of NBG set theory) and the maps between two objects form a set.
Now, it is well known that the presheaves does not form a (locally small) category anymore: the maps between two objects does not form a set, and even the objects does not form a class anymore.
Instead, one can consider the small presheaves, which it is known to be a locally small category: the homorphisms between to small presheaves form a set clearly.
But, how it is proved that the objects form a class?
I understand that the "collection" of objects is some sort of colimit of classes, indexed on a class. But what sort of axiom one uses in order to know one gets a class?
Recall that (in NBG) the elements of any class are sets. In particular, it is impossible to have any class of presheaves when every presheaf itself is a proper class. Similarly, it is impossible to have any class of presheaf morphisms when every presheaf morphism itself is a proper class. The trick is to use proxies for the objects and morphisms – which implicitly means replacing the purported category with an equivalent one.
For the objects, instead of small presheaves, we take pairs $(\mathcal{I}, X)$ where $\mathcal{I}$ is a small category and $X$ is a diagram $\mathcal{I} \to \mathcal{C}$. (For the purpose of this exercise, the domain and codomain are not considered part of the data of a diagram; so $X$ is just an ordered pair whose first component is a set of ordered pairs consisting of an object of $\mathcal{I}$ and an object of $\mathcal{C}$ and whose second component is etc.) We define morphisms $(\mathcal{I}, X) \to (\mathcal{J}, Y)$ to be elements of the following set: $$\textstyle \varprojlim_{\mathcal{I}^\textrm{op}} \varinjlim_\mathcal{J} \mathcal{C} (X, Y)$$ Concretely, that means that a morphism $f : (\mathcal{I}, X) \to (\mathcal{J}, Y)$ consists of, for each object $i$ in $\mathcal{I}$, an object $j_i$ in $\mathcal{J}$ and a morphism $f_i : X (i) \to Y (j_i)$ in $\mathcal{C}$, such that certain equations induced by the diagram $X : \mathcal{I} \to \mathcal{C}$ are satisfied modulo certain equivalence relations induced by the diagram $Y : \mathcal{J} \to \mathcal{C}$. Composition is defined in the obvious way (but checking that it is well defined is tedious).
It is clear by construction that the above is a locally small category... but is it equivalent to the intuitive category of small presheaves? Well, a small presheaf is a small colimit of representables, say $\varinjlim_\mathcal{I} h_X$. Given $\varinjlim_\mathcal{I} h_X$ and $\varinjlim_\mathcal{J} h_Y$, we have $$\begin{aligned} \textstyle \textrm{Hom} \left( \varinjlim_\mathcal{I} h_X, \varinjlim_\mathcal{J} h_Y \right) & \textstyle \cong \varprojlim_{\mathcal{I}^\textrm{op}} \textrm{Hom} \left( h_X, \varinjlim_\mathcal{J} h_Y \right) \\ & \textstyle \cong \varprojlim_{\mathcal{I}^\textrm{op}} \varinjlim_\mathcal{J} h_Y (X) \\ & \textstyle = \varprojlim_{\mathcal{I}^\textrm{op}} \varinjlim_\mathcal{J} \mathcal{C} (X, Y) \end{aligned}$$ by the Yoneda lemma. It is then clear that we have a fully faithful and essentially surjective on objects functor from the locally small category defined above to the intuitive category of small presheaves.