Yoneda Lemma says that, for a locally small category $\mathcal{C}$, an object $A$ in $\mathcal{C}$ and a functor $F:\mathcal{C}^{op}\to \textbf{Set}$, the natural transformations $[Hom(-,A):F]$ is in one-to-one correspondence with elements of $F(A)$.
My question is the following:
- $\textit{A priori}$, we do not know that the collection $[Hom(-,A):F]$ is. Now we show a "bijection" to $F(A)$. How does it imply that the collection $[Hom(-,A):F]$ is a set.
If the above question does not make sense, then I try to frame it in another fashion.
For doing mathematics we need to consider primitive undefined objects and their "properties". Mathematicians argue that foundations can be built in many ways like-
- sets and $\in$ (Well, wasn't this the plan)
sets and $f$ (function) (Leinster's $\textit{Basic Category Theory}$)
types (friend told this to me)
So what are the objects we are considering in Category theory. If we take an arbitrary collection of morphisms, when can we say it is a set or not. If it is not a set, how is its properties different from that of a set.
If this is vague question, specfically considering Yoneda Lemma, how can we say that $[Hom(-,A):F]$ is a set.