Number of Natural Transformation

Question

I am working on the following problem:

Let $F:\text{Set}\to\text{Set}$ be the functor that has the object map $X\to X\times X$ and the morphism map $(f:X\to Y)\to (f\times f:X\times X\to Y\times Y)$ where $(f\times f)(a,b)= (f(a),f(b))$.
Find the number of natural transformations from $F$ to itself.

I think I may be able to find a representation for the functor $F$, but I actually have made no progress from there. Thanks in advance!

Answer 1

Hint 1. Recall that the Yoneda Lemma states that for functors $F : \mathcal{C} \to \mathbf{Set}$ we have a bijection $F(X) \to \mathrm{Hom}(\mathrm{Hom}(X,-),F)$, $u \mapsto (f \mapsto F(f)(u))$.

Hint 2. Notice that the functor $F : \mathbf{Set} \to \mathbf{Set}$, $X \mapsto X \times X$ is isomorphic to the functor $\mathrm{Hom}(\{1,2\},-)$.

Hint 3. You will find that there are exactly four morphisms $F \to F$. You can write them down also in a very concrete way, just using the definition of the bijection above.