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We introduce a new method for precisely relating certain kinds of algebraicstructures in a presheaf category and judgements of its internal type theory.The method provides a systematic way to organise complex diagrammatic reasoningand generalises the well-known Kripke-Joyal forcing for logic. As anapplication, we prove several properties of algebraic weak factorisationsystems considered in Homotopy Type Theory.

Kripke-Joyal forcing for type theory and uniform fibrations