Representable functors

Which pair $(A, u)$ is a universal element of the forgetful functor $U : \cat{Ab} \to \cat{Set}$ from abelian groups to sets?

The group $\Z^2 = \langle a, b \mid ab = ba \rangle$ represents the functor sending a group $G$ to the set of commuting pairs $\{(x, y) \in G^2 \mid xy = yx\}$. How many group homomorphisms $\Z^2 \to S_3$ are there?

How many different natural isomorphisms $\Hom(\Z, -) \Rightarrow U$ are there, where $U : \cat{Grp} \to \cat{Set}$ is the forgetful functor?

Which pairs $(A, u)$ with $u \subseteq A$ are universal elements of the contravariant power-set functor $\mathcal{P} : \cat{Set}^{\mathrm{op}} \to \cat{Set}$ (acting by preimage)? Choose all that apply.

In the proof of Theorem 2 we have $\varphi : A \to A'$ with $F(\varphi)(u) = u'$ and $\psi : A' \to A$ with $F(\psi)(u') = u$. Why is $\psi \circ \varphi = \id_A$?

Which of these functors $\cat{Set} \to \cat{Set}$ is not representable?

Let $\mathcal{F} : \cat{Top}^{\mathrm{op}} \to \cat{Set}$ send a space to its set of closed subsets (a continuous map acts by preimage). Which pair represents $\mathcal{F}$? Here $\mathbb{S} = \{0, 1\}$ is the Sierpiński space with open sets $\varnothing, \{1\}, \{0, 1\}$.

In the lesson's Lean file, imagePowerset : Type ⥤ Type is the covariant power-set functor (direct images). Which Lean statement says that it is not representable, i.e. that no type $A$ represents it?

Since $\Z/n = \langle a \mid a^n \rangle$ represents $G \mapsto \{g \in G \mid g^n = 1\}$, the number of homomorphisms $\Z/n \to \Z/m$ equals the number of $k \in \Z/m$ with $nk \equiv 0 \pmod m$. Write hom_count(n, m) computing this number (for $n, m \ge 1$). Then compare your answers with $\gcd(n, m)$.