I'm reading Soergel's "Derivierte Kategorien und Funktoren" [1]. In 1.2.6, Soergel defines a localization functor as follows. Let $F: \mathcal{C} \to \mathcal{D}$ be a functor and denote by $S$ the set (ignoring set-theoretical issues that may arise) of morphisms of $\mathcal{C}$ that are taken to isomorphisms in $\mathcal{D}$. The universal property of localization yields a unique functor $\tilde{F}: \mathcal{C}_S \to \mathcal{D}$ through which $F$ factors. (The indexed $S$ denotes the localization at $S$.) Then $F$ is called a localization functor if $\tilde{F}$ is an equivalence of categories.
He gives the following example of a localization functor. Let $\mathcal{K}$ denote the quiver $\bullet \to \bullet$ and denote by $\operatorname{Rep}(\mathcal{K})$ the category of quiver representations in abelian groups. Consider the functor $F: \operatorname{Rep}(\mathcal{K}) \to \mathcal{Ab}$ into the category of abelian groups that projects a quiver representation $V_1 \to V_2$ to $V_1$. It is then claimed that $F$ is a localization functor.
For me it is easy to see that $\tilde{F}$ is essentially surjective (it is, in fact, surjective on objects) and full. However, I am struggling to show the faithfullness of $\tilde{F}$. At this point I should mention that Soergel constructs localizations from the free category on a certain quiver $\mathcal{C} \coprod S^{-1}$ by factoring out an equivalence relation, as in the nlab article under "General construction" (see [2]). So showing faithfulness of $\tilde{F}$ amounts to showing the following:
For a morphism $f$ of quiver representations, let $f^{(1)}$ denote the map in the first component. Consider two strings $g_1 \circ \overline{s_1} \circ \ldots \circ g_n \circ \overline{s_n}$ and $h_1 \circ \overline{t_1} \circ \ldots \circ h_m \circ \overline{t_m}$ in the free quiver on $\operatorname{Rep}(\mathcal{K})\coprod S^{-1}$ such that $$g_1^{(1)} \circ (s_1^{(1)})^{-1} \circ \ldots \circ g_n^{(1)} \circ (s_n^{(1)})^{-1} = h_1^{(1)} \circ (t_1^{(1)})^{-1} \circ \ldots \circ h_m^{(1)} \circ (t_m^{(1)})^{-1}.$$ Show that the two strings have the same image in $\operatorname{Rep}(\mathcal{K})_S$ (i.e. that they are related via the equivalence relation). This is where I got stuck - any help would be greatly appreciated.
[1] http://home.mathematik.uni-freiburg.de/soergel/Skripten/XXTD.pdf
[2] https://ncatlab.org/nlab/show/localization
