  This paper solves the first problem of the open problems in topos theory
posted by William Lawvere, which asks the existence of a Grothendieck topos
that has a proper class many quotient topoi. This paper concretely constructs
such Grothendieck topoi, including the presheaf topos of the free monoid
generated by countably infinite elements $\mathbf{PSh}(M_\omega)$. Utilizing
the combinatorics of the classifying topos of the theory of inhabited objects
and considering pairing functions, the problem is reduced to making rigid
relational structures. This is accomplished by using Kunen's theorem on
elementary embeddings in set theory.
