The class $l(F)$ of morphisms which have the left lifting property with respect to $F$ is stable under transfinite compositions. I am reading Cisinski's Higher Categories and Homotopical Algebra and I am having trouble trying to verify some claims there. My background in category theory is not very solid. I would like some help to understand and prove one of that claims.

Let $i: A \to B$ and $p: X \to Y$ be two morphisms in a category
$\mathcal{C}$. We say that $i$ has the left lifting property with
respect to $p$, or, equivalently, that $p$ has the right lifting
property with respect to $i$, if any commutative square of the form

has a diagonal filler

(i.e. a morphism $h$ such that $hi = a$ and $ph = b$).
A class of morphisms $F$, in a category $\mathcal{C}$, is stable under
transfinite compositions if, for any well ordered set $I$, with initial element $0$, for any functor $X: I \to \mathcal{C}$ such that,
for any element $i \in I$, $i \neq 0$, the colimit $\varinjlim_{j < i} X(j)$ is representable and the induced map
$$\varinjlim_{j < i} X(j) \to X(i)$$ belongs to the class $F$, the colimit $\varinjlim_{i \in I} X(i)$ exists and the canonical morphism $X(0) \to \varinjlim_{i \in I} X(i)$ belongs to $F$ as well.
Proposition: Let $\mathcal{C}$ be a category with a class of morphisms $F$. Then the class $l(F)$ is stable under transfinite compositions.

The above proposition is just a small part of the Proposition 2.1.4 in the book. I was able to prove the other parts, so I am not completely lost. The problem is in this definition of transfinite composition... I know what a colimit is, I know what a well ordered set is, but I am not sure I understood the definition at all. Where does the condition "the colimit $\varinjlim_{j < i} X(j)$ is representable" comes into play? What is the canonical morphism $X(0) \to \varinjlim_{i \in I} X(i)$? Is this one of that morphisms in the definition of cocone? How does the above definition is equivallent to the nLab definition of transfinite composition? How to prove the above Proposition?
 A: Cisinski means that the colimit exists, in more common terminology for English-speaking mathematicians. (In the Grothendieck school, it is traditional to speak of a colimit as a presheaf, so that it always exists, and the question is whether this presheaf is representable in the starting category.)
Now, as for what the condition says: first, consider the case that $i=i'+1$ is a successor ordinal. Then $\mathrm{colim}_{j<i}X(j)=X(i'),$ and so the condition simply says that the map $X(i')\to X(i)$ is in $F.$ The case in which $i$ is a limit ordinal is the natural way to say what is morally the same thing: for instance, the map $\mathrm{colim}_{i\in\omega} X(i)\to X(\omega)$ is in $F.$ If you're homotopy-theoretically minded you might picture this as the union of a countable chain of cofibrations, all mapping into $X(\omega).$
Yes, the canonical morphism from $X(0)$ to the colimit must be the appropriate leg of the colimiting cocone.
With all this spelled out, the proposition should be easy to prove by transfinite induction.
