Assuming Choice, a small category with products or coproducts indexed by the cardinality of its morphism set is a preorder

Suppose distinct parallel arrows $f,g:A\rightrightarrows B$ exist, and let $P=B^{\kappa }$ be a product of the constant $\kappa$-family. For each subset $S\subseteq \kappa$, [F1] gives a unique $h_S:A\to P$ whose $i$th projection is $f$ when $i\in S$ and $g$ when $i\mathrm{\notin }S$.

If $S\mathrm{\ne }T$, choose $i$ in their symmetric difference. The $i$th composites of $h_S$ and $h_T$ are $f$ and $g$ in some order, so $h_S\mathrm{\ne }{h}_T$. Thus $S\mapsto h_S$ injects $\mathcal{P}(\kappa )$ into $\mathrm{Mor}(\mathcal{C})$.

By [F2], the codomain has cardinality $\kappa$, whereas [L1] says the domain has strictly larger cardinality. This contradiction proves that no distinct parallel arrows exist. Identities and composition already make the object relation reflexive and transitive, so [F3] makes $\mathcal{C}$ a preorder.

Apply [L2] to ${\mathcal{C}}^{\mathrm{o}\mathrm{p}}$. Its morphism set has the same cardinality, a $\kappa$-indexed coproduct in $\mathcal{C}$ is a product there, and being a preorder is unchanged by reversal. This proves the coproduct clause.