  Presheaf models of dependent type theory have been successfully applied to
model HoTT, parametricity, and directed, guarded and nominal type theory. There
has been considerable interest in internalizing aspects of these presheaf
models, either to make the resulting language more expressive, or in order to
carry out further reasoning internally, allowing greater abstraction and
sometimes automated verification. While the constructions of presheaf models
largely follow a common pattern, approaches towards internalization do not.
Throughout the literature, various internal presheaf operators ($\surd$,
$\Phi/\mathsf{extent}$, $\Psi/\mathsf{Gel}$, $\mathsf{Glue}$, $\mathsf{Weld}$,
$\mathsf{mill}$, the strictness axiom and locally fresh names) can be found and
little is known about their relative expressivenes. Moreover, some of these
require that variables whose type is a shape (representable presheaf, e.g. an
interval) be used affinely. We propose a novel type former, the transpension
type, which is right adjoint to universal quantification over a shape. Its
structure resembles a dependent version of the suspension type in HoTT. We give
general typing rules and a presheaf semantics in terms of base category
functors dubbed multipliers. Structural rules for shape variables and certain
aspects of the transpension type depend on characteristics of the multiplier.
We demonstrate how the transpension type and the strictness axiom can be
combined to implement all and improve some of the aforementioned
internalization operators (without formal claim in the case of locally fresh
names).
