Let \(U\) be a universe. Then \(\mathbb {A}\) represents the functor \(\mathord {\textnormal {\textsf {Sh}}}:{(\mathord {\textnormal {\textsf {Topos}}}(U))}^{\mathord {\textnormal {\textsf {op}}}}\rightarrow \mathord {\textnormal {\textsf {Logos}}}(U)\). That is, we have a natural equivalence \(\mathord {\textnormal {\textsf {Hom}}}_{\mathord {\textnormal {\textsf {Topos}}}(U)}(X,\mathbb {A})\simeq \mathord {\textnormal {\textsf {Sh}}}(X)\).