I am trying to understand the characterization of the class of closed embeddings into a normal Hausdorff space as the class of continuous maps satisfying the left lifting property with respect to a unique map, presented in https://ncatlab.org/nlab/show/colimits+of+normal+spaces. The last change made in this page is anonymous but I have found a paper not giving more details in https://mishap.sdf.org/.

First of all, if the rule is $a<b$ if and only if $b$ is in the closure of $a$, the poset given in the nLab page is the wrong one: it should be $a>b<c>d<e$ because the closed points are $a,c,e$ (there is also a typo in the description of the closed points) and the points $b,d$ are open.

For the author of this note (I guess M. Gavrilovich), a continuous map is a closed embedding into a normal Hausdorff space if and only if it satisfies the LLP with respect to the map $$ g:\{a>b<c>d<e\} \longrightarrow \{a>b=c=d<e\} $$

Could someone explain what seems to be evident for the author of this note please ?

What I can understand is that a map $\varnothing\to X$ satisfies the LLP with respect to $g$ if and only if $X$ is normal (but not necessarily Hausdorff). Indeed, assume the LLP. Take two disjoint closed subsets $A$ and $E$ of $X$ taken to the closed points $a$ and $e$ respectively of $\{a>b=c=d<e\}$; the other points of $X$ are taken to the open point $b=c=d$. Then the existence of the lift $\ell:X\to \{a>b<c>d<e\}$ provides two closed subsets $F_1=\ell^{-1}(\{a,b,c\})$ and $F_2=\ell^{-1}(\{c,d,e\})$. Then $F_1^c \cap F_2^c=\varnothing$. And $E\subset F_1^c$ and $A\subset F_2^c$. Hence $X$ is normal. Conversely, if $X$ is normal, then the LLP is satisfied.

I do not understand either why the Tietze extension theorem is mentioned which is a characterization of normal Hausdorff spaces.