Proposition 31, p. 26 — monos and epis in DSpace are the injective and surjective maps

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 map $f:X\to Y$ of $D$ spaces, with underlying function $\underline f=f_1:\underline X\to\underline Y$,

This identifies the monomorphisms and epimorphisms of $\mathrm{DSpace}$ concretely and is used to analyse strong monomorphisms.

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 $(T, h_T)$. Mono and epi are taken in the category of $D$ spaces (a full subcategory of presheaves); $\underline f$ is the component of the natural transformation $f$ at $T$.