In the proof of the Yoneda lemma, Leinster states this (p. 97):
(The hat is the function $\alpha\mapsto \alpha_A(1_A)$ and the tilde is its inverse.)
But I don't understand how he applies Lemma 1.3.11 above.
When applied to the Yoneda lemma, Lemma 1.3.11 says, as far as I understand, that a natural transformation $$\alpha:[\mathscr A^{op},\textbf{Set}](H_{\star},-)\implies -(\star)$$ is a natural isomorphism iff $$\alpha_{(A,X)}:[\mathscr A^{op},\textbf{Set}](H_A,X)\to A(X)$$ is an isomorphism (= bijection in this case).
But it doesn't tell anything about naturality, does it? What am I missing?
And I'm not sure I even understand why "in principle we have to prove naturality of both $(\hat)$ and $(\tilde)$". According to the lemma, if we know $\alpha_{(A,X)}$ is a bijection and that $\alpha$ is a natural transformation (that is, $(\hat)$ is natural), then the result will follow. Or am I wrong here?
Yes, it does : Suppose you manage to prove that the hat transformation is natural.
Then, since you already know that its components are isomorphisms (I assume that was proved earlier on), you know that the hat transformation is a natural isomorphism, hence it has an inverse as a natural transformation.
But an inverse as a natural transformation must have its components be the inverses of the components, i.e. it has to be the tilde transformation.
So if you prove that the hat transformation is natural, so is its inverse : the tilde transformation.