Universal properties

Let $|X| = 2$, $|A| = 2$ and $|B| = 3$. How many functions $X \to A \times B$ are there? (Use the universal property: such a function is the same as a pair of functions $X \to A$, $X \to B$.)

The free monoid on a one-element set $\{a\}$ is isomorphic to which monoid?

Let $\bar f$ be the unique monoid homomorphism from the free monoid on $\{a, b\}$ to $(\mathbb{Z}, +, 0)$ with $\bar f(a) = 2$ and $\bar f(b) = -1$. What is $\bar f(abbab)$?

Let $A, B$ be non-empty sets and $P = A \times B \times \{0, 1\}$, with $p(a, b, i) = a$ and $q(a, b, i) = b$. Which part of the universal property of the product does $(P, p, q)$ fail?

In the proof that two universal arrows $(F, \eta)$ and $(F', \eta')$ are isomorphic, we know $U(\psi \circ \varphi) \circ \eta = \eta$. What exactly lets us conclude $\psi \circ \varphi = \mathrm{id}_F$?

Regard the positive integers ordered by divisibility ($a \to b$ when $a \mid b$) as a category. Which statements are true? Choose all that apply.

In the Lean structure UniversalArrow X U, with fields obj, unit : X ⟶ U.obj obj and lift, which field states the uniqueness clause? (Recall unit ≫ U.map g means $U(g) \circ \eta$.)

An equivalence relation on a finite set is given by its list of classes, e.g. [{0, 3}, {1, 4}, {2, 5}], and a function by a dict. Write descend(f, classes) that returns the induced function $\bar f$ on classes — a dict from each class (as a frozenset) to a value, with $\bar f([x]) = f(x)$ — or None if $f$ is not constant on some class (so that no $\bar f$ with $\bar f \circ q = f$ exists).