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)?