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We show that certain diagrams of $\infty$-logoses are reconstructed ininternal languages of their oplax limits via lex, accessible modalities, whichenables us to use plain homotopy type theory to reason about not only a single$\infty$-logos but also a diagram of $\infty$-logoses. This also provides ahigher dimensional version of Sterling's synthetic Tait computability -- a typetheory for higher dimensional logical relations. To prove the main result, weestablish a precise correspondence between the lex, accessible localizations ofan $\infty$-logos and the lex, accessible modalities in the internal languageof the $\infty$-logos. To do this, we also partly develop the Kripke-Joyalsemantics of homotopy type theory in $\infty$-logoses.

Homotopy type theory as internal languages of diagrams of $\infty$-logoses