In a locally small category, each object represents a presheaf (and dually a copresheaf ). The Yoneda lemma is the natural bijection
for every presheaf . A natural family out of is therefore no more and no less than an element of . The Yoneda embedding is consequently full and faithful, so an object is determined exactly by the maps into it. Thus sits inside its presheaf category , which is its free cocompletion, and iff . The lemma powers representability, the optic representation theorem (double-Yoneda), and the category-of-elements correspondence.
References
Saunders Mac Lane, Categories for the Working Mathematician (1971), III.2 (Yoneda lemma)