$\textbf{5.2.} \enspace \triangleright$ Prove that $\emptyset$ is the unique initial object in $\mathsf{Set}. \enspace [\S5.1]$

---

$\textbf{5.3.} \enspace \triangleright$ Prove that the final objects are unique up to isomorphism. $[\S5.1]$

---

$\textbf{5.5.} \enspace \triangleright$ What are the final objects in the category considered in $\S 5.3\text{?} \> [\S 5.3]$

Solution

Recall that the morphisms in $\S 5.3$ are commutative diagrams

---

$\textbf{5.6} \enspace \triangleright$ Consider the category corresponding to endowing (as in Example 3.3) the set $\Z^+$ of positive integers with the divisibility relation. Thus there is exactly one morphism $d \to m$ in this category if and only if $d$ divides $m$ without remainder; there is no morphism between $d$ and $m$ otherwise. Show that this category has products and coproducts. What are their 'conventional' names? $[\S\text{VII}.5.1]$

$$\\$$