We present cubical-categorical-logic, a library of formalized category theory in Cubical Agda. The library’s core idea is to treat syntax as a free categorical structure whose dependent eliminator is stated via (displayed) universal properties. From the same reusable components we have proven canonicity and conservativity results across several type theories, as well as the coherence theorem for monoidal categories. In this talk, we discuss these applications and reflect on Cubical Agda as a host for mechanized metatheory, where it is sometimes a boon and sometimes a bane.