$\def\colim{\operatorname{colim}}
\def\hom{\operatorname{Hom}}$Let $\mathcal{C}$ be a category and let $S$ be a class of morphisms of $\mathcal{C}$ that is a [left multiplicative system][1]. 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][2]). 

> **Fact.** $S^{-1}\mathcal{C}$ is not locally small in general [[ref][3]].

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][4]. 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][5] 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][6]. So [05Q2][4] 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][7] and [Verdier's thesis][8], Corollaire 2.2.4.


  [1]: https://stacks.math.columbia.edu/tag/04VC
  [2]: https://stacks.math.columbia.edu/tag/04VB
  [3]: https://mathoverflow.net/q/321141/101848
  [4]: https://stacks.math.columbia.edu/tag/05Q2
  [5]: https://stacks.math.columbia.edu/tag/05Q0
  [6]: https://stacks.math.columbia.edu/tag/0013
  [7]: https://ncatlab.org/nlab/show/calculus+of+fractions#construction_of_the_localization
  [8]: http://www.numdam.org/item/AST_1996__239__R1_0.pdf