Bill Lawvere’s definition of an atomic infinitesimal space is as an object $\Delta$ in a topos $\mathcal{T}$ such that the inner hom functor $(-)^\Delta : \mathcal{T} \to \mathcal{T}$ has a right adjoint (is an atomic object).
Notice that by definition of inner hom, $(-)^\Delta$ always has a left adjoint. A right adjoint can only exist for very particular objects. Therefore the term amazing right adjoint
Assume $\mathcal{T} = Sh(C)$ is a Grothendieck topos, that the Grothendieck topology on the site $C$ is subcanonical. Let $\Delta \in C \hookrightarrow Sh(C)$ be a representable object.
Then $(-)^\Delta$ has a right adjoint, hence $\Delta$ is an atomic infinitesimal space, precisely if it preserves colimits.
This is a special case of the general adjoint functor theorem.
For if $(-)^\Delta$ preserves colimits, its right adjoint is
The $Y_\Delta$ defined this way is indeed a sheaf, due to the assumption that $(-)^\Delta$ preserves colimits. So this is indeed a right adjoint.
Ieke Moerdijk, Gonzalo Reyes, appendix 4 of, Models for Smooth Infinitesimal Analysis