The particular Yoneda end and the enriched functor category have different size requirements

The enriched end in Strong enriched Yoneda lemma as a particular end is a single object with a universal enriched wedge. By contrast, Functor category $[\mathcal{C},\mathcal{D}]$ forms an ordinary category of set-coded functors and ordinary natural transformations when its source is small; it neither defines enriched hom-objects nor imposes completeness hypotheses on the base.

When the enriched functor category $[\mathcal{A},\mathcal{V}]$ is formed, its hom-object between two enriched functors $F$ and $G$ is an enriched end of the objects $[FA,GA]$. Forming the whole enriched functor category therefore asks for such an end for every pair $(F,G)$, whereas the strong Yoneda theorem directly exhibits one particular end whether or not all of those other ends exist.

So two claims must be kept separate:

• the weak and strong enriched Yoneda lemmas identify a specific object or set attached to a single representable functor;
• the existence of the entire enriched functor category requires all of its enriched hom-objects to exist, commonly under additional smallness and completeness hypotheses.

This page proves only the former. Even for the set-object sources used here, one particular Yoneda end does not by itself construct every hom-object of a full enriched functor category.