Adjunctions

For the free group functor $F : \cat{Set} \to \cat{Grp}$ and the forgetful functor $U : \cat{Grp} \to \cat{Set}$, which statement is correct?

Let $|A| = 2$, $|B| = 3$ and $|C| = 2$. How many functions $A \to C^B$ are there (where $C^B$ is the set of functions $B \to C$)? Check your answer against $|\Hom(A \times B, C)|$.

On $(\Z, \le)$ let $l(n) = 2n$. Its right adjoint $u$ satisfies $2n \le m \iff n \le u(m)$ for all integers $n, m$. What is $u(7)$? (And what is $u(-7)$, for yourself?)

Given an adjunction $\varphi : \Hom(FA, B) \cong \Hom(A, GB)$, the unit is $\eta_A = \varphi(\mathrm{id}_{FA})$. Which formula gives the counit $\varepsilon_B : FGB \to B$?

Why does the forgetful functor $U : \cat{Field} \to \cat{Set}$ have no left adjoint?

Abelianisation $G \mapsto G^{\mathrm{ab}}$ is left adjoint to the inclusion $\iota : \cat{Ab} \hookrightarrow \cat{Grp}$. Which statements follow from “right adjoints preserve limits, left adjoints preserve colimits”? Select all that apply.

In proving that unit and counit with the triangle identities give an adjunction, we show $\varepsilon_B \circ F(Gk \circ \eta_A) = k$ for $k : FA \to B$. The computation is $\varepsilon_B \circ FGk \circ F\eta_A = k \circ \varepsilon_{FA} \circ F\eta_A = k$. Which facts justify the two equalities, in order?

Mathlib defines GaloisConnection l u for maps l : α → β and u : β → α between preorders. Which proposition is it?

Write right_adjoint(l, P, Q, le_p, le_q) for a monotone map l (a dict) from a finite poset P to a finite poset Q. Return the dict u with l[a] ≤ b ⟺ a ≤ u[b] if it exists, using u[b] = max{a : l[a] ≤ b}, and None otherwise (including when that maximum does not exist).