  The Carath\'eodory extension theorem is a fundamental result in measure
theory. Often we do not know what a general measurable subset looks like. The
Carath\'eodory 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 Carath\'eodory
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 Carath\'eodory extension theorem.
