A complete locally small category with a jointly weakly initial set has an initial object, without class-indexed choice

Regard $\mathcal{S}$ as the full subcategory it spans. Its objects form a set, and by [L2] the union of the hom-sets between them is a set, so this full subcategory is small. By [L1] its inclusion has a limiting cone $(L,p_S:L\to S{)}_{S\in \mathcal{S}}$. This remains valid when $\mathcal{S}$ is empty: then joint weak initiality implies that $\mathcal{C}$ has no objects, so the theorem's hypotheses cannot hold for a category with a target object.

Fix one target $C$. Joint weak initiality supplies some $S_0\in \mathcal{S}$ and one map $h:S_0\to C$, so $h\circ p_{S_0}:L\to C$ exists. The witness is chosen only for this fixed target, not simultaneously for a proper class of targets; hence $L$ is weakly initial.

By [L2] the collection $\mathcal{C}(L,L)$ is a set, so the one-object category whose arrows are the endomorphisms of $L$ is small, and sending its object to $L$ and each arrow to itself is a diagram. By [L1] that diagram has a limit; write its single leg as $j:I\to L$. The cone condition says exactly that $\alpha \circ j=j$ for every $\alpha \in \mathcal{C}(L,L)$, and $j$ is monic, since $j\circ x=j\circ y$ makes $x$ and $y$ both mediate the same cone, so the uniqueness clause of [L4] gives $x=y$.

$I$ is weakly initial: for a target $C$, step 2.1 supplies a map $L\to C$ and composing it with $j$ gives $I\to C$. Again one witness is used for one fixed target.

Let $f,g:I\rightrightarrows C$ and let $m:K\to I$ be their equalizer, which exists by [L1] and is monic by [L6]; it satisfies $fm=gm$ by [L5]. Step 2.1 gives $u:L\to K$, so $j\circ m\circ u:L\to L$ is an endomorphism of $L$ and step 2.2 gives $(j\circ m\circ u)\circ j=j$. Rewriting the left side as $j\circ (m\circ u\circ j)$ and cancelling the monomorphism $j$ by [L7] yields $m\circ u\circ j=1_I$. Hence $m$ is a split epimorphism as well as monic, so $m\circ (u\circ j\circ m)=(m\circ u\circ j)\circ m=m=m\circ 1_K$ and left-cancelling $m$ gives $u\circ j\circ m=1_K$; thus $m$ is an isomorphism by [L7]. From $fm=gm$ and the invertibility of $m$ we get $f=g$. There is therefore exactly one morphism $I\to C$ for every target $C$, so $I$ is an initial object of $\mathcal{C}$.