Examples of mathematical statements made with adjoint functors I am wondering if it is possible to use the adjoint functors in topos theory for statements in analysis.
Any examples would be warmly welcomed. Though I would prefer simpler, atomic, lemmas or theorems if possible.
Thanks for your time.
Edit 1: Examples do not have to be restricted to analysis, I just thought those might be nice.
Edit 2: At this point even a comment would be nice.
Edit 3: To clarify, what I am looking for are theorems stated or proved using the adjoint functors relating to predicate calculus.

In this paper you will find  some "applications" of category theory to functional analysis:
www.eweb.unex.es/eweb/extracta/Vol-25-2/25J2Castillo.pdf
You will find more in its bibliography.
