Studying localizations of categories from Weibel, I came across the notion of a locally small multiplicative system S (see Weibel 10.3.6 'Set-Theoretic Considerations'). Assuming that S is locally small a functor written as $\text{Hom}_C(-,Y)$ from $S_X$ to $Sets$ is constructed. On objects it sends an $s \in S$ to the set of all fractions $fs^{-1}$. I do not understand why the collection of all fractions $fs^{-1}$ is supposed to be a set considering that the collection of all $f$ (i.e. $\text{Hom}_C(X_1,Y)$) is not required to be a set.
Edit: In Weibel's book every category is locally small. So everything is clear now.
Below is the related part in Weibel: "A multiplicative system $S$ is called locally small (on the left) if for each $S$ there exists a set $S_X$ of morphisms in $S$, all having target $X$, such that for every $X_1 \to X$ in S there is a map $X_2 \to X_i$ in $C$ so that the composite $X_2 \to X_i \to X$ is in $S_X$. If $S$ is locally small, then $\text{Hom}_S(X, Y)$ is a set for every $X$ and $Y$. To see this, we make $S_X$ the objects of a small category, a morphism from $X_1 \to X$ to $t: X_2 \to X$ being a map $X_2 \to X_1$ in $C$ so that $t$ is $X_2 \to X_1 \to X$. The Ore condition says that by enlarging $S_X$ slightly we can make it a filtered category (2.6.13). There is a functor $\text{Hom}_C(—, Y)$ from $S_X$ to $Sets$ sending $s$ to the set of all fractions $fs^{-1}$ , and $\text{Hom}_S(X, Y)$ is the colimit of this functor."