typical contexts
For $\Gamma : \mathcal{E} \to \mathcal{B}$ a functor we say that it has codiscrete objects if it has a full and faithful right adjoint $coDisc : \mathcal{B} \hookrightarrow \mathcal{E}$.
An object in the essential image of $coDisc$ is called a codiscrete object.
This is for instance the case for the global section geometric morphism of a local topos $(Disc \dashv \Gamma \dashv coDisc) \mathcal{E} \to \mathcal{B}$.
If one thinks of $\mathcal{E}$ as a category of spaces, then the codiscrete objects are called codiscrete spaces.
The dual notion is that of discrete objects.
$\Gamma$ is a faithful functor on morphisms whose codomain is concrete.
If $\mathcal{E}$ has a terminal object that is preserved by $\Gamma$, then $\mathcal{E}$ has concrete objects.
This is (Shulman, theorem 1).
If $\mathcal{E}$ has codiscrete objects and has pullbacks that are preserved by $\Gamma$ and , then $\Gamma$ is a Grothendieck fibration.
This is (Shulman, theorem 2).
(shape modality $\dashv$ flat modality $\dashv$ sharp modality)
$(ʃ \dashv \flat \dashv \sharp )$
(reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)
$(Red \dashv ʃ_{inf} \dashv \flat_{inf})$