Lemma 2.2 (Yoneda). Assuming that:

• $C$ is a locally small category

• $A\in \mathrm{ob}C$

• $F:C\to Set$ a functor

• (i) There is a bijection between natural transformations $C(A,\bullet )\to F$ and elements of $FA$.
• (ii) Moreover, this bijection is natural in $A$ and $F$.