TITLE: Examples of Kan extensions, adjunctions, and (co)monads in analysis, Lie theory, and differential geometry?
QUESTION [26 upvotes]: In introductory texts on category theory, it seems like the majority of examples come from algebraic topology, algebra, and logic. 
What are some good examples of Kan extensions, adjunctions, and (co)monads in analysis, Lie theory, and differential geometry? 
Since limits and colimits can be characterized as Kan extensions or adjunctions, we have the obvious standard constructions: (co)products, (co)equalizers, etc., but these are common to a lot of categories we work with. Are there any examples more specific to analysis and differential geometry?
(Crosspost from Math.SE)

REPLY [3 votes]: The Schwartz kernel theorem, e.g, that all continuous linear maps from the space $\cal S(\mathbb R^m)$ of Schwartz functions on $\mathbb R^m$ to tempered distributions $\cal S'(\mathbb R^n)$ on $\mathbb R^n$ are given by "kernels" $K(,)$ in $\cal S'(\mathbb R^{m+n})$, is ${\rm Hom}(\cal S\otimes \cal S,\mathbb C)\approx {\rm Hom}(\cal S, \cal S')$, assuming/when the tensor product exists in the category of locally convex topological vector spaces. (Emphatically, the usual "projective" and/or "injective" tensor products are not tensor products in the full categorical sense.) This adjunction is an instance of the Cartan-Eilenberg ${\rm Hom}(A\otimes B,C)\approx {\rm Hom}(A,{\rm Hom}(B,C))$. Existence of the genuine tensor product uses the nuclearity of $\cal S$, that is, the demonstrable fact that $\cal S$ is a countable projective limit of Hilbert spaces with transition maps that are Hilbert-Schmidt. (And "trace-class" might best be characterized as being the composition of two Hilbert-Schmidt operators.)