Products by maps: a first universal property

Let $X = \{1, 2, 3\}$, $f : X \to \{\text{red}, \text{blue}\}$ with $f(1) = \text{red}$ and $f(2) = f(3) = \text{blue}$, and $g : X \to \Z$ with $g(n) = n^2$. What is $\langle f, g \rangle(2)$?

Let $|X| = 2$, $|A| = 2$ and $|B| = 3$. How many functions $X \to A \times B$ are there?

Let $A = B = \{0, 1\}$. Which of these $(P, p, q)$ are products of $A$ and $B$? (Choose all that apply.)

In the proof of Theorem 3 we showed that $\psi \circ \varphi = \mathrm{id}_P$. Which fact justifies that final equality?

Let $P = A \times B \times \{0, 1\}$, with $p(a, b, c) = a$ and $q(a, b, c) = b$. Let $X$ have $3$ elements and fix $f : X \to A$ and $g : X \to B$. How many functions $h : X \to P$ satisfy $p \circ h = f$ and $q \circ h = g$?

Given p : P → A, q : P → B, f : X → A and g : X → B, which Lean proposition says "there is exactly one $h$ compatible with $f$ and $g$"?

Represent a finite function as a Python dict. Write pairing(f, g), which takes two dicts with the same keys and returns the dict $x \mapsto (f(x), g(x))$, and components(h), which takes a dict whose values are pairs and returns the two dicts $(\pi_A \circ h, \pi_B \circ h)$. Check that the two are inverse to each other (Theorem 2).

Let $|A| = 3$ and $|B| = 4$, and suppose a set $P$ with maps $p : P \to A$ and $q : P \to B$ has the universal property of the product. How many elements must $P$ have?

Make the positive integers into a category: there is one arrow $x \to y$ exactly when $x$ divides $y$ (and no arrow otherwise). A product of $12$ and $18$ is a number $P$ with arrows $P \to 12$ and $P \to 18$ such that every $X$ with arrows $X \to 12$ and $X \to 18$ has an arrow $X \to P$ (uniqueness is automatic, as there is at most one arrow between two numbers). What is $P$?