Since the OP explicitly asked for examples in harmonic analysis and there are not many answers touching upon this, I'd like to point out that harmonic analysis does make heavy use of categorical notions in the representation theory-heavy branches of the field. For example, the Poisson transform is a crucial tool at the heart of harmonic analysis, and in order to work with it, the following two concepts are useful, for example:
Induction of smooth representations of Lie groups and induction of $(\mathfrak g,K)$-modules, with the important special case of parabolic induction. This is a prototypical example of an adjunction. Here "ordinary" categories are sometimes not enough but (if one wants to be formal) one requires enriched categories to take into account that the $\mathrm{Hom}$-spaces involved in the Frobenius reciprocity isomorphisms are equipped with operator topologies.
The Casselman-Wallach theorem: For every reductive pair $(G,K)$ the restriction functor to the $K$-finite vectors is an equivalence between the category of smooth admissible Fréchet representations of moderate growth with continuous linear $G$-maps as morphisms and the category of Harish-Chandra $(\mathfrak g,K)$-modules with $(\mathfrak g,K)$-homomorphisms as morphisms.