The Carath'eodory extension theorem and the Kan extension of strict transformations

A categorical proof of the Carathéodory extension theorem

We develop a general framework for extensions of transformations between poset-valued functors and give several results on the existence and construction of extensions of these transformations.We prove that (co)lax and strict transformations represent (pre)measures and outer measures by certain (co)lax and strict transformations.We then show that transformations and functors corresponding to measures satisfy these results, which proves the carath\'eorody extensions theorem.