Higher categories

A bicategory with exactly one object $\star$ is the same thing as which of the following?

How many ways are there to fully bracket a composite of six composable 1-cells $f_6 f_5 f_4 f_3 f_2 f_1$? (These are the vertices of the associahedron $K_6$.)

Let $\alpha : F \Rightarrow G$ (functors $\mathcal{C} \to \mathcal{D}$) and $\gamma : I \Rightarrow J$ (functors $\mathcal{D} \to \mathcal{E}$). Which formulas give the component $(\gamma \ast \alpha)_X : IFX \to JGX$ of the horizontal composite? (Choose all that apply.)

In the proof of the interchange law $(\delta \cdot \gamma) \ast (\beta \cdot \alpha) = (\delta \ast \beta) \cdot (\gamma \ast \alpha)$ in $\mathbf{Cat}$, after expanding components the two sides are $K(\beta_X) K(\alpha_X) \delta_{FX} \gamma_{FX}$ and $K(\beta_X)\, \delta_{GX} J(\alpha_X)\, \gamma_{FX}$. Which fact finishes the proof?

The Eckmann–Hilton argument assumes that both operations are unital. Which pair of operations on a set with at least two elements satisfies the interchange law $(a \star b) \odot (c \star d) = (a \odot c) \star (b \odot d)$ but is not commutative, showing the unit hypothesis cannot be dropped?

Let B be a bicategory in Lean ([Bicategory B]) and a : B. Which statement formalises Corollary 3, 'the 2-endomorphisms of an identity 1-morphism commute'?

Which of these are strict 2-categories, with composition of 1-cells associative and unital on the nose? (Choose all that apply.)

Which statement about coherence in dimension 3 is correct?

Write count_bracketings(n), the number of full bracketings of a word of n letters (use the recursion over the outermost bracket), and count_moves(n), the number of edges of the associahedron $K_n$: pairs of bracketings that differ by one associator. Use the fact that every bracketing of $n \ge 2$ letters has exactly $n - 2$ internal brackets, each allowing exactly one associator move.