= Slice-small presheaf construction
{title2=$(G^F)(A)=\operatorname{Nat}(F\circ\mathrm{dom},G\circ\mathrm{dom})_{\mathcal C/A}$}
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.
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