Products, coproducts, initial and terminal objects

In $\cat{Set}$, which statement is correct?

Which of these categories have a zero object (an object that is both initial and terminal)? Select all that apply.

Regard the positive integers, ordered by divisibility ($a \le b$ iff $a \mid b$), as a category. What is the coproduct of $12$ and $18$?

In $\cat{Set}$, let $|X| = 2$, $|A| = 2$ and $|B| = 3$. How many morphisms $X \to A \times B$ are there? (Use the universal property rather than listing them.)

In the proof that two products $(P, \pi_i)$ and $(Q, \rho_i)$ of $A$ and $B$ are isomorphic, we have $\varphi : P \to Q$ and $\psi : Q \to P$ with $\rho_i \varphi = \pi_i$ and $\pi_i \psi = \rho_i$. Why is $\psi \circ \varphi = \mathrm{id}_P$?

What is the coproduct of $\Z/2$ and $\Z/3$ in the category of commutative rings with identity?

Which statements are true? Select all that apply.

Mathlib writes h ≫ π₁ for $\pi_1 \circ h$ (first h, then π₁). Which Lean field correctly expresses the uniqueness clause of the universal property of a product with projections π₁ : P ⟶ A, π₂ : P ⟶ B?

Write glb(elements, leq, a, b) returning the product of a and b in a finite poset — the greatest lower bound — or None if it does not exist. Here elements is a list and leq(x, y) decides $x \le y$.