Slice-small presheaf construction ($(G^F)(A)=\mathrm{Nat}(F\circ \mathrm{dom},G\circ \mathrm{dom}{)}_{\mathcal{C}/A}$)

If every slice category $\mathcal{C}/A$ is essentially small, the displayed exponential object and the set of sieves on a category at each $A$ are small even when $\mathcal{C}$ is large. Together with pointwise finite limits, they give the elementary-topos constructions in a universe admitting the large diagrams. This does not force the entire functor category to be locally small in the original universe: a large discrete category has terminal slices, but maps between its constant-one and constant-two presheaves are arbitrary class-indexed binary families.