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Section “Necessity of naturality” sounded like there is nothing interesting going on if we look at the unnatural isomorphism of hom-sets. But this isn’t the case, already in the 60s a graph-theoretical and purely categorical facts of this form have been established. I’m adding the two earliest references.
Tomas Jakl
The entry fell short of really citing the following in full, so I have added it now:
Yoshiki Kinoshita, Nobuo Yoneda (1930-1996)
&
Saunders MacLane, The Yoneda Lemma,
Math. Japonica, 47, No. 1 (1998) 155-156
(pdf, also: posting to catlist in 1996)
This edit needs analogous adjustments to those pointed out in another thread here.
I am out of energy for the moment. Please look into it. Generally, please try to keep in mind that nLab entries need to retain a minimum of logical consistency as you drop material into them that is copied from elsewhere.
This happens when different authors edit different bits at different times.
If you have the energy to harmonize the notation, it would be a welcome service to the community.
The notation $C(...)$ tends to indicate that the construction also works in the enriched context, while the notation $Hom_C(...)$ makes clear that one does insist to mean plain Hom-sets.
So I suppose that $C(...)$ is a good choice for this page.