90% visual proof of Contravariant Yoneda Lemma
By the fact that \alpha_X : \text{Hom}_C(X,X) \to AX we have that \alpha_X(\text{id}_X) =: u \in AX and we're done with mapping any natural map \alpha : \text{Hom}_C(\cdot, Y) \to A to an element u \in AX.
The sides of the triangular prism in the bottom diagram need to commute for each f:Y\to X, g : Z \to Y in C. For one, the triangular endcaps must commute because A, \text{Hom}_C(\cdot, X) are both contravariant functors. Next, the three square sides commute by naturality. We must then have that any time \alpha_Y(f) = A(f)\circ u, the diagram commutes.
Comments
Post a Comment