Proposition 29, p. 25 — every representable presheaf is a D space

Let $D$ be a small category with a coverage $K$ and a terminal object $T$ (written $1$ in the paper), and suppose $(D,K,T)$ is a concrete site: representables are sheaves, $\operatorname{hom}(1,-)$ is faithful, and covering families are jointly surjective on points. Write $\underline{X}=X(1)$ for the underlying set of a presheaf $X$ and $\underline{\varphi}:\underline{D}\to\underline{X}$, $d\mapsto X(d)(\varphi)$, for the underlying function of a plot $\varphi\in X(D)$.

For every object $X$ of $D$, the representable presheaf $\operatorname{hom}(-,X):D^{\mathrm{op}}\to\mathrm{Set}$ is a $D$ space:

The result embeds the site into its category of generalized spaces via the Yoneda embedding.

Formalization Note The site is small ($D$ in universe $u$, presheaves valued in $\mathrm{Type}_u$), covering families are presieves, and the terminal domain is passed explicitly as $(T, h_T)$. The proposition's second sentence, that the underlying set of $\operatorname{hom}(-,X)$ is $\underline X=\operatorname{hom}(1,X)$, holds by definition in Lean (the value of the Yoneda presheaf at $T$ is the hom-set $T\to X$) and is not restated.