I do not understand here what do the words colimit of a filtered diagram in $\cal P$,not just in $\cal F$ on the 2nd page mean, namely the word just there; is some direction trivial ?
Namely,in general if one has categories $\cal K\subseteq L$ and a scheme $F:\cal D\to K$ then is it always the case that the colimit
in $\cal K$ is a colimit in $\cal L$ or vice versa ? I cannot calculate this as in $\cal L$ we have more commutative co-cones and also
morphisms from the colimit cocone so we may violate both the uniqueness and existence of
the colimit induced morphism in $\cal K$: $\operatorname{colim} F$.
BTW, what is bigger $\cal P$ or $\cal F$ in that paper cited ?
My guess is $$\cal P\subseteq F.$$
For those who cannot open the file the two relevant pages:
Best Answer
A flat object in a variety $\mathcal{V}$ is a colimit in $\mathcal{V}$ of some small filtered diagram of finitely presentable projective objects in $\mathcal{V}$. Note that every object in $\mathcal{V}$ is a colimit in $\mathcal{V}$ of some small filtered diagram of finitely presentable objects in $\mathcal{V}$ – the requirement of projectivity is doing all the work in this definition. Every finitely presentable projective object is flat. (All of this will make more sense if you think about the concrete example of the category of $R$-modules for some ring $R$.)
It is the case that if $\mathcal{K}$ is a full subcategory of $\mathcal{L}$ and you have a cocone in $\mathcal{K}$ that is a colimit cocone in $\mathcal{L}$, then it is also a colimit cocone in $\mathcal{K}$. The converse is false, i.e. the inclusion $\mathcal{K} \hookrightarrow \mathcal{L}$ may fail to preserve colimits. This is why it is important to specify not just whether a colimit is in the subcategory or not, but also whether their universal property is with respect to the subcategory or the ambient category.