Yoneda lemma

Yoneda embedding

The Yoneda embedding is an embedding of a locally small category into the Category of presheaves, that

• maps an object to the presheaf
• maps a morphism to the natural transformation , whose components are the pushforwards of

Proof of embedding

Let . By the Yoneda lemma we have

where using the notation of the ^proof given and

so , implying

so in particular, is fully faithful. is also clearly injective on objects, since if then

so .

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