Franklin's Notes

Yoneda Functor

Given a locally small category $\mathsf C$, we may define the Yoneda Functor $Y:\mathsf C^\text{op}\to \mathsf{Set}^{\mathsf C}$ which acts on objects by mapping each $x\in\mathsf C$ to the representable hom-functor $\mathsf C(x,-)$, and acts on morphisms by sending each morphism $f_{\text{op}}:y\to x$ of $\mathsf C^\text{op}$ (corresponding to the morphism $f:x\to y$ of $\mathsf C$) to the natural transformation $\mathsf C(y,-)\to\mathsf C(x,-)$ defined by pre-composition by $f$. As a consequence of the Yoneda lemma , for any category $\mathsf C$, the functor $Y$ is full and faithful .



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