# Why isn’t the localization $C[W^{-1}]$ (locally) small when $C$ is small and $W$ admits a calculus of (right) fractions?

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)?