We describe the categorical semantics for a simply typed variant and a
simplified dependently typed variant of Cocon, a contextual modal type theory
where the box modality mediates between the weak function space that is used to
represent higher-order abstract syntax (HOAS) trees and the strong function
space that describes (recursive) computations about them. What makes Cocon
different from standard type theories is the presence of first-class contexts
and contextual objects to describe syntax trees that are closed with respect to
a given context of assumptions. Following M. Hofmann's work, we use a presheaf
model to characterise HOAS trees. Surprisingly, this model already provides the
necessary structure to also model Cocon. In particular, we can capture the
contextual objects of Cocon using a comonad ♭ that restricts presheaves
to their closed elements. This gives a simple semantic characterisation of the
invariants of contextual types (e.g. substitution invariance) and identifies
Cocon as a type-theoretic syntax of presheaf models. We further extend this
characterisation to dependent types using categories with families and show
that we can model a fragment of Cocon without recursor in the Fitch-style
dependent modal type theory presented by Birkedal et. al.