Synthetic topos theory
[004E] Exercise

Let \(U\) be a universe, let \(V\) be a universe strictly greater than \(U\), and let \(X\) be a \(U\)-topos. Then we have a canonical equivalence

\(\mathord {\textnormal {\textsf {Sh}}}(\mathord {\textnormal {\textsf {Topos}}}^{(1)}(U),V)/X\simeq \mathord {\textnormal {\textsf {Hom}}}_{\mathord {\textnormal {\textsf {Sh}}}(X)}(\mathord {\textnormal {\textsf {1}}},(\mathord {\textnormal {\textsf {Sh}}}(\mathord {\textnormal {\textsf {Topos}}}^{(1)}(U),V))\mathrel {@}X)\)