The Yoneda lemma

In the Yoneda lemma $\mathrm{Nat}(\Hom(A, -), F) \cong F(A)$, which element of $F(A)$ corresponds to a natural transformation $\alpha : \Hom(A, -) \Rightarrow F$?

A directed graph $G$ (allowing loops and parallel edges) has $7$ vertices and $12$ edges. View it as a functor on the category $\mathbb{E}$ with arrows $s, t : E \to V$. How many graph homomorphisms are there from the walking edge $\Hom(E, -)$ — one edge from a vertex $s$ to a vertex $t$ — to $G$?

On the category $\cat{Ab}$ of abelian groups, $\Hom(\Z/3, -)$ sends $G$ to its elements of order dividing $3$, and $\Hom(\Z, -)$ sends $G$ to all of its elements. How many natural transformations $\Hom(\Z/3, -) \Rightarrow \Hom(\Z, -)$ are there?

Which of these rules define natural transformations $U \Rightarrow U$, where $U : \cat{Grp} \to \cat{Set}$ is the forgetful functor (a rule must be defined for every group and commute with every homomorphism)? Choose all that apply.

In the proof that $\Psi(\Phi(\alpha)) = \alpha$ we write $\alpha_X(f) = \alpha_X(f \circ \id_A) = F(f)(\alpha_A(\id_A))$. Which fact justifies the second equality?

Which of these conclusions are justified for objects $A, B$ of a locally small category? Choose all that apply.

Let $G$ be a group. Which permutations $\sigma$ of the set $G$ satisfy $\sigma(xh) = \sigma(x)h$ for all $x, h \in G$?

Mathlib's coyoneda.obj (op A) is the functor $\Hom(A, -)$. Which Lean statement formalises "if $\Hom(A, -) \cong \Hom(B, -)$ naturally, then $A \cong B$"?

The cyclic group $\Z/n$ acts on $X = \{0, \dots, m-1\}$ when its generator acts by a permutation p (a list with p[x] the image of x) satisfying $p^n = \mathrm{id}$. An equivariant map from the regular action is a function a : Z/n → X (a list of length n) with a[(k + 1) % n] == p[a[k]] for all k. Write count_equivariant(n, p) that counts such maps by brute force over all functions (use itertools.product). The Yoneda lemma predicts the answer len(p).