To prove the Yoneda Lemma one defines a bijection between $[\mathcal{A}^{op},\mathbf{Set}](\hom(-,A),X)$ and $X(A)$ and shows that this bijection is natural in $A$ and $X$.
In my textbook this bijection is $\alpha\mapsto \alpha_A(\text{id}_A)$ which was my first thought as well. However, there is the question how we could have defined it alternatively.
So my question is: which alternative is meant and how does one get this idea?
Leinster writes: "How else could we possibly define it?"
This is not understood as an exercise to find another definition.
This is understood in the following sense: There is no real other way to define it, right? You have only one choice.
A german translation would be: "Wie sollte man das auch sonst definieren?"