  Let $X$ be a complete measure space of finite measure. The Lebesgue transform
of an integrable function $f$ on $X$ encodes the collection of all the
mean-values of $f$ on all measurable subsets of $X$ of positive measure. In the
problem of the differentiation of integrals, one seeks to recapture $f$ from
its Lebesgue transform. In previous work we showed that, in all known results,
$f$ may be recaputed from its Lebesgue transform by means of a limiting process
associated to an appropriate family of filters defined on the collection of all
measurable subsets of $X$ of positive measure. The first result of the present
work is that the existence of such a limiting process is equivalent to the
existence of a Von Neumann-Maharam lifting of $X$.
  In the second result of this work we provide an independent argument that
shows that the recourse to filters is a \textit{necessary consequence} of the
requirement that the process of recapturing $f$ from its mean-values is
associated to a \textit{natural transformation}, in the sense of category
theory. This result essentially follows from the Yoneda lemma. As far as we
know, this is the first instance of a significant interaction between category
theory and the problem of the differentiation of integrals.
  In the Appendix we have proved, in a precise sense, that \textit{natural
transformations fall within the general concept of homomorphism}. As far as we
know, this is a novel conclusion: Although it is often said that natural
transformations are homomorphisms of functors, this statement appears to be
presented as a mere analogy, not in a precise technical sense. In order to
achieve this result, we had to bring to the foreground a notion that is
implicit in the subject but has remained hidden in the background, i.e., that
of \textit{partial magma}.
