To begin, let me first state some definitions from Joy of Cats,
Definition 1. Let $(\mathbf{A},U)$ be a concrete category over $\mathbf{X}$ (i.e, where $\mathbf{A}$ is a category and $U:\mathbf{A}\to\mathbf{X}$ is a faithful functor).
An $\mathbf{A}$-morphism $A\overset{f}{\to} B$ is called initial provided that for any $\mathbf{A}$-object $C$ and for the $\mathbf{X}$-morphism $U(C)\overset{g}{\to} U(A)$ there exists a unique $\mathbf{A}$-morphism $C\overset{g^\ast}{\to} A$ such that $U(g^\ast)=g$ whenever for the $\mathbf{X}$-morphism $U(C) \overset{U(f)\circ g}{\longrightarrow} U(B)$ there exists a unique $\mathbf{A}$-morphism $C\overset{h^\ast}{\to} B$ such that $U(h^\ast)=U(f)\circ g$.
An initial morphism $A\overset{f}{\to} B$ for which $U(f)$ is a monomorphism is called an embedding.
Definition 2. Let $(\mathbf{A},U)$ over $\mathbf{X}$ be a concrete category and $C$ be an $\mathbf{A}$-object. Then $C$ will be said to be injective provided that for any embedding $A\overset{m}{\longrightarrow} B$ and any $\mathbf{A}$-morphism $A\overset{f}{\longrightarrow}C$ there exists an $\mathbf{A}$-morphism $B\overset{g}{\longrightarrow} C$ such that $f=g\circ m$.
Definition 3. Let $(\mathbf{A},U)$ over $\mathbf{X}$ be a concrete category and $A,B$ be two $\mathbf{A}$-objects. An embedding $A\overset{m}{\longrightarrow}B$ is said to be essential if for any $\mathbf{A}$-object $C$ and any $\mathbf{A}$-morphism $B\overset{f}{\longrightarrow}C$ whenever $f\circ m$ is an embedding, so is $f$.
My Question
I am trying to understand the reason behind calling these embeddings as stated in Def. 3 "essential". I have already tried to discuss this question here and although the term "essential submodule" has been beautifully motivated by Denis Nardin (as you can see here), I couldn't find motivation the term "essential embedding" in a similar way. So my question is,
Why are these embeddings as elaborated in Def. 3 called essential embeddings?