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Weak pushout is weak initial object

Let $(\mathcal{C},\cup, \cap, \emptyset)$ be a poset, $X$ be a nonempty set and $\mathcal{S}$ be a set of subobjects of the free cocompletion of $X$, that is, $\mathcal{S}$ is closed under finite coproducts.
A collection of subobjects $D\subseteq \mathcal{S}$ is called a directed system if $f:A\to B$ in $\mathcal{S}$ implies that $f:A\cup X\to B\cup X$ factors through a unique $g:A\cup X\to B$ in $\mathcal{S}$.
Let $\varphi :\mathcal{S}\to \mathcal{C}$ be a functor, and let $K$ be the directed system of subobjects of the free cocompletion of $X$ defined by $\varphi$.
I want to show that a $Y\in \mathcal{C}$ is a weak initial object of $\mathcal{C}$ with respect to $\varphi$ iff it is the weak pushout of some directed system $D\subseteq \mathcal{S}$ with respect to the embedding $\varphi$ (if I understand this correctly).
I need help and advice to prove this for weak initial object, and I don’t know how to proceed for weak pushout.
Thank you for your help.

A:

Weak pushout is usually taken as a derived notion; you’re right that it can be defined in the adjoint category. Here

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