Can everything in category theory be restated in a sufficiently expressive type theory?
I've been reading the homotopy type theory (HoTT) book as well as some articles on ncat lab
It seems that HoTT can be interpreted as the syntax for an -groupoid or more generally an (,1)-category. That article and related articles on ncatlab seem to suggest that within an arbitrary category C the objects can be translated as types (as in HoTT types) and morphisms can be translated as dependent types. Thus a category can be interpreted as an "abstract data structure" and functors between categories are just ordinary (type theoretic) functions between data types/structures in type theory.
Is that translation correct, and if so, does that suggest that any categorical notion has a simple reformulation (without much work) to a homotopy type theoretic form? Coming from a computational background, for some reason, just replacing the word "category" with "abstract data structure" and a few other terms in category theory with type theoretic terminology seems to make category theory much more intuitive than it appears.