= Solution
An <initial object> by itself is a <weakly initial set>. For the converse, let $(W_i)_{i\in I}$ be a small weakly initial family and form its <product in a category> $W$. Given $X$, choose an arrow $W_i\to X$; its composite with the projection $W\to W_i$ shows that $W$ is weakly initial.
Local smallness makes $\operatorname{End}(W)$ a <set>, so completeness supplies the simultaneous <equalizer> $e:E\to W$ of all endomorphisms of $W$ with $1_W$. Thus $he=e$ for every $h\in\operatorname{End}(W)$. The object $E$ is weakly initial because it maps to $W$.
For parallel arrows $a,b:E\to X$, take their <equalizer> $j:Y\to E$. Weak initiality of $W$ supplies $t:W\to Y$. The endomorphism $ejt$ of $W$ satisfies $(ejt)e=e$, and monicity of $e$ gives $jte=1_E$. Therefore $j$ is both a <monomorphism> and a <split epimorphism>, hence an <isomorphism>. Since $aj=bj$, we obtain $a=b$. There is already at least one arrow from $E$ to every $X$, so $E$ is initial. This proves the <initial-object lemma for complete categories with a weakly initial set>, with smallness used exactly where the endomorphisms are equalized.
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