Lemma 4.3.5 (Yoneda lemma). Let $U, V \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. Given any morphism of functors $s : h_ U \to h_ V$ there is a unique morphism $\phi : U \to V$ such that $h(\phi ) = s$. In other words the functor $h$ is fully faithful. More generally, given any contravariant functor $F$ and any object $U$ of $\mathcal{C}$ we have a natural bijection

Appeared in some form in [Yoneda-homology]. Used by Grothendieck in a generalized form in [Gr-II].

**Proof.**
For the first statement, just take $\phi = s_ U(\text{id}_ U) \in \mathop{Mor}\nolimits _\mathcal {C}(U, V)$. For the second statement, given $\xi \in F(U)$ define $s$ by $s_ V : h_ U(V) \to F(V)$ by sending the element $f : V \to U$ of $h_ U(V) = \mathop{Mor}\nolimits _\mathcal {C}(V, U)$ to $F(f)(\xi )$.
$\square$

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