  We present new induction principles for the syntax of dependent type
theories, which we call relative induction principles. The result of the
induction principle relative to a functor F into the syntax is stable over the
codomain of F. We rely on the internal language of presheaf categories. In
order to combine the internal languages of multiple presheaf categories, we use
Dependent Right Adjoints and Multimodal Type Theory. Categorical gluing is used
to prove these induction principles, but it not visible in their statements,
which involve a notion of model without context extensions. As example
applications of these induction principles, we give short and boilerplate-free
proofs of canonicity and normalization for some small type theories, and sketch
proofs of other metatheoretic results.
