= Solution
A <weakly initial set> is a set of objects $(W_i)_{i\in I}$ such that every object receives at least one map from some $W_i$. An <initial object> is itself a singleton <weakly initial set>. We prove the converse without replacing uniqueness by weak initiality.
Let $\mathcal E$ be a <complete category> that is <locally small>. Form $W=\prod_i W_i$. For every object $A$, choose $W_i\to A$ and compose with the projection $W\to W_i$, so $W$ is weakly initial. Completeness supplies a <terminal object>, so the given <weakly initial set> cannot be empty.
The <endomorphisms> of $W$ form a <set>. Take a simultaneous <equalizer> $e:I\to W$ of all of them with $1_W$; equivalently, equalize the two maps from $W$ into $\prod_{s\in\mathcal E(W,W)}W$ with components $s$ and $1_W$. Thus $se=e$ for every endomorphism $s$. The object $I$ is weakly initial, since $I\xrightarrow eW\to A$ exists for every $A$. Weak initiality of $W$ gives a map $u:W\to I$. Applying the equalizing identity to $eu$ yields $eue=e$, so monicity of $e$ gives $ue=1_I$.
For any <endomorphism> $v:I\to I$, the map $evu$ is an <endomorphism> of $W$. Therefore $evue=e$. Since $ue=1_I$, this becomes $ev=e$, giving $v=1_I$. Now let $f,g:I\rightrightarrows A$ and form their <equalizer> $j:E\to I$. Weak initiality of $I$ gives $h:I\to E$. The composite $jh$ is an <endomorphism> of $I$, hence the identity. Consequently $f=fjh=gjh=g$.
There is at least one map from $I$ to every object, and the argument proves that there is at most one. \b[The object $I$ is an <initial object>.] This establishes the <initial-object lemma for complete categories with a weakly initial set> using only small <products in a category> and <equalizers>.
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