In the proof of the full exact embedding theorem of the "Handbook of Categorical Algebra 2" of F. Borceux at the step 2, there is something I do not understand. It is at page 82 just after the colimits of the previous equalizer diagram, he says :



> 
> Since $\downarrow \delta\_A$ is an initial part of $\mathcal{D}$
> 
> 
> 


I tried to prove it but I can't. Any help? Moreover, how did he use this fact to apply Proposition 2.11.2 of Vol.1 ? Indeed, this proposition works for limits and initial part. Here we have a colimit, so we should have a terminal part to apply it, and $\downarrow \delta\_A$ is not terminal. Am I right?


EDIT : my idea is that it is a colimit on a contravariant functor, and thus the proposition applies with an inital part.


To be clear, I understand "initial part" as "the inclusion is a final functor". And I understand "final functor" in the sense of the volume 1, which the dual version of what we find on [ncatlab](https://ncatlab.org/nlab/show/final+functor).



Misunderstanding of the proof of the Embedding Theorem in Borceux
Sorry, it was not that complicated : it comes from the fact that $ \mathcal{D}$ is a $\wedge$-semilattice. Thus for $D \in \mathcal{D}$ we just have to compute $D'=\delta\_A \wedge D$ to have an element in $\downarrow \delta\_A$ such that $D' \leq D$.


