In the presence of a calculus of (right) fractions, one may prove that every equivalence class of the general localization—the quotient of $ F(UC +_{obj W} W^{op})$ , the free category on the amalgamation of the underlying quiver of $ C$ and $ W^{op}$ with the objects of $ W$ , tautologously inverting $ W$ —is represented by a span $ (~\overset{w}{\leftarrow}~\overset{\varphi}{\rightarrow}~)$ with $ w \in W$ and $ \varphi \in C$ . Further, composition by exchange of `interior’ cospan for span is associative up to the equivalence. And best of all, the general equivalence relation may be substituted for a much simpler one: $ $ (~\overset{w}{\leftarrow}~\overset{\varphi}{\rightarrow}~) \sim (~\overset{v}{\leftarrow}~\overset{\psi}{\rightarrow}~) \iff \exists s,t \in C:~ ws=vt \in W,~ \varphi s=\psi t $ $ This is all pretty cool and gives a much more tractable presentation of the localization. However, insofar as these $ \sim$ -classes of spans between $ c$ and $ d$ are expressible as: $ $ C[W^{-1}](c,d) \cong \operatorname{colim}\limits_{(\circ \overset{w}{\to} c)\in W} C(\circ,d) $ $ where the colimit is taken over the full subcategory of $ C/c$ whose objects are morphisms from $ W$ , I don’t see how the localization can fail to be (locally) small. $ \operatorname{Set}$ is cocomplete and the overcategory $ C/c$ is small.

Would appreciate someone pointing me toward my error(s)?