A remark on Yoneda's Lemma

TL;DR

This paper discusses a specific aspect of Yoneda's Lemma, highlighting the canonical natural transformation from a representable functor to a subset functor of another functor, emphasizing the collection of natural transformations.

Contribution

It provides a new perspective on Yoneda's Lemma by identifying a canonical natural transformation related to subset functors and natural transformations.

Findings

Identifies a canonical natural transformation from a representable functor to a subset functor.

Highlights the role of natural transformations in the context of Yoneda's Lemma.

Provides insight into the structure of natural transformations in category theory.

Abstract

Yoneda'e Lemma is about the canonical isomorphism of all the natural transformations from a given representable covariant (contravariant, reps.) functor (from a locally small category to the category of sets) to a covariant (contravariant, reps.) functor. In this note we point out that given any representable functor and any functor we have the canonical natural transformation from the given representable functor to the "subset" functor of the given functor, "collecting all the natural transformations".