The Carathéodory extension theorem is a fundamental result in measure theory. Often we do not know what a general measurable subset looks like. The Carathéodory extension theorem states that to define a measure we only need to assign values to subsets in a generating Boolean algebra. To prove this result categorically, we represent (pre)measures and outer measures by certain (co)lax and strict transformations. The Carathéodory extension then corresponds to a Kan extension of strict transformations. 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 proceed by showing that transformations and functors corresponding to measures satisfy these results, which proves the Carathéodory extension theorem.