Natural transformations

Let $F, G : \mathcal{C} \to \mathcal{D}$ be functors and $\alpha_A : F(A) \to G(A)$ a family of morphisms. Which equation must hold for every $f : A \to B$ for $\alpha$ to be natural?

Let $T : \cat{Set} \to \cat{Set}$ be the functor $T(X) = X \times X \times X$, $T(f) = f \times f \times f$. How many natural transformations $T \Rightarrow \mathrm{id}_{\cat{Set}}$ are there?

Which of these families are natural transformations (between the functors indicated)? Select all that apply.

Let $\alpha : \mathrm{id}_{\cat{Ab}} \Rightarrow \mathrm{id}_{\cat{Ab}}$ be a natural transformation with $\alpha_{\Z}(1) = 7$. What is $\alpha_{\Z/12}(5)$? Give the representative in $\{0, 1, \dots, 11\}$.

In the proof that an equivalence $F$ (with $\eta : \mathrm{id} \cong GF$, $\varepsilon : FG \cong \mathrm{id}$) is full, we set $f = \eta_B^{-1} \circ G(h) \circ \eta_A$ for a given $h : F(A) \to F(B)$ and showed $GF(f) = G(h)$. What is needed to conclude $F(f) = h$?

Which statement about $\mathrm{ev}_V : V \to V^{**}$ on the category of all vector spaces over a field $k$ is true?

Mathlib writes composition in diagrammatic order: f ≫ g means “first f, then g”. For α : F ⟶ G between functors C ⥤ D and f : X ⟶ Y in C, which Lean statement is the naturality square?

Which pairs of categories are equivalent? Select all that apply.

A binary operation on $X = \{0, \dots, n-1\}$ is stored as a tuple t with t[a * n + b] $= \alpha(a, b)$. Write natural_tables(n) returning the list of all such tables that commute with every function $f : X \to X$, i.e. t[f[a] * n + f[b]] == f[t[a * n + b]] for all $a, b$ (with f a tuple of values). Theorem 3 predicts the answer.