The theory of signatures, is a small type theory. It is given as a QIIT, and kaposi2019-constructing-qiit show that if a type theory supports this QIIT, then it supports all finitary QIITs.
This is a restriction of theory of codes from kaposi2017-qiit-codes focused on targeting finitary QIITs, as opposed to all QIITs.
References
From kaposi2019-constructing-qiit. See also damato2025-formalizing-containers.