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)?



Why isn't the localization $C[W^{-1}]$ (locally) small when $C$ is small and $W$ admits a calculus of (right) fractions?
If $C$ is small, $C[W^{-1}]$ is again small, even without a calculus of fractions. The problem in general is that when $C$ is a locally small large category, the zig-zag sequences can range over finite sequences of objects of $C$, which form a large set.


Essentially, the problem is that the hom-sets in the localization before taking quotients can, without additional assumptions, have the cardinality of the set of objects, and there may be sets of morphisms even after quotienting with cardinality equal to the cardinality of C.


This is a problem when C is large, since it means that the hom-sets can be large.


