$\def\colim{\operatorname{colim}} \def\hom{\operatorname{Hom}}$Let $\mathcal{C}$ be a category and let $S$ be a class of morphisms of $\mathcal{C}$ that is a left multiplicative system. The localization of $\mathcal{C}$ with respect to $S$ is a category $S^{-1}\mathcal{C}$ with same objects as $\mathcal{C}$ and where a morphism $X\to Y$ in $S^{-1}\mathcal{C}$ is a pair of maps $X\xrightarrow{f}Y'\xleftarrow{s}Y$ in $\mathcal{C}$, with $s\in S$, modulo a certain equivalence relation (cf. 04VB).
Fact. $S^{-1}\mathcal{C}$ is not locally small in general (even when $\mathcal{C}$ is) [ref].
The localization $S^{-1}\mathcal{C}$ comes equipped with a canonical localization functor $Q:\mathcal{C}\to S^{-1}\mathcal{C}$. In Gabriel, Zisman, Calculus of Fractions and Homotopy Theory, Proposition I.3.1 states that $Q$ preserves finite direct limits. Actually, their proof works for finite colimits, as it is stated and done with the same proof in the Stacks Project 05Q2. The proof boils down to noticing $$ \label{hom_loc}\tag{1} \operatorname{Hom}_{S^{-1}\mathcal{C}}(X,Y)=\underset{(Y\to Y')\in Y/S}{\colim}\hom_\mathcal{C}(X,Y') $$ (see 05Q0 for the definition of $Y/S$) and then using that
$(*)$ finite limits commute with filtered colimits in the category of sets.
Here's the big but: in the colimit \eqref{hom_loc} the category $Y/S$ is a not small category in general, so we cannot apply $(*)$ to something that may not be a set! The Stacks Project has a funny way of dealing with this: for them, by default all categories are small (unless one works with certain specific categories) 0013. So 05Q2 constitutes a sound proof, for one assumes $\mathcal{C}$ is small. The problem is Gabriel and Zisman's proof: they do it for a possibly non-small category, so I don't see how one can get something meaningful now.
Even though it is not clear in which category the colimit is taken over, formula \eqref{hom_loc} shows up in the nLab and Verdier's thesis, Corollaire 2.2.4.
