Limits and colimits

Which shape $\mathcal{J}$ has the pullback as the limit of a diagram of that shape?

Let $A = \{1, \dots, 6\}$, $B = \{1, 2, 3, 4\}$, $C = \{0, 1, 2\}$, and $f : A \to C$, $g : B \to C$ both be “remainder modulo 3”. How many elements does the pullback $A \times_C B$ have in $\cat{Set}$?

In $\cat{Ab}$, what is the equaliser of $f, g : \Z \to \Z/6$ given by $f(n) = n \bmod 6$ and $g(n) = 3n \bmod 6$?

In the proof that products and equalisers give all limits, the limit of $D$ is the equaliser of two maps $s, t : \prod_j D(j) \rightrightarrows \prod_{u : i \to j} D(j)$. What is the $u$-component of $t$ (for $u : i \to j$)?

Which of these functors preserve all limits? Select all that apply.

In $\Z_2 = \varprojlim \Z/2^n$, the element $\tfrac13$ is a compatible tower $(x_1, x_2, x_3, \dots)$ with $3x_n \equiv 1 \pmod{2^n}$. What is $x_5 \in \{0, 1, \dots, 31\}$?

Regard $(\Q, \le)$ as a category. The increasing sequence $3 \le 3.1 \le 3.14 \le 3.141 \le \cdots$ of decimal truncations of $\pi$ is a diagram of shape $(\N, \le)$. What is its colimit?

Which Lean statement expresses the universal property of the equaliser $\{x // f x = g x\}$ of f g : X → Y in the category of types?

Write pullback(A, B, f, g) returning the pullback of f : A → C and g : B → C in finite sets, as the list of pairs (a, b) with f[a] == g[b] (here A, B are lists and f, g dicts). Then pullback_size(A, B, f, g) should return its size computed fibre by fibre, without forming the pairs.