Initial-object lemma for complete categories with a weakly initial set

A locally small category with all small categorical limits and a small weakly initial set has an initial object. Take the product $W$ of the weakly initial family, then the simultaneous equalizer $e:E\to W$ of all endomorphisms of $W$ and its identity. For any parallel $a,b:E\to X$, their equalizer $j:Y\to E$ receives a map $t:W\to Y$. The equation $(ejt)e=e$ forces $jte=1_E$, so $j$ is invertible and $a=b$. Weak initiality supplies existence of maps from $E$. This is the smallness mechanism in the Freyd general adjoint functor theorem.