category theory - How to understand quasi-inverse of a function f∘g∘f = f?

Recently I was studying the quasi-inverse. Before I studied the quasi-inverse, I revisited the inverse and the left-right inverse.

inverse function:

Let $f : X → Y$, $g : Y → X$ is inverse of $f$, if only if, $f∘g = id_{Y}$ and $g∘f = id_{X}$.

It is easy to understand.

right-inverse function:

Let $f : X → Y$, $g : Y → X$ is right-inverse of $f$ (or section of $f$ ), if only if , $f∘g = id_{Y}$.

It means that $f$ must be surjective and $g$ must be injective. It is also very intuitive.

Now I start to study quasi-inverse:

One thing I have to explain here is that the "quasi-inverse" does not seem to be a precise terminology and I can't find any information about quasi-inverse in wikipedia or nlab. (I study it because the form of "quasi-inverse" appears in many branches of mathematics, e.g. in category theory, adjoint functors needs to satisfy triangular identity. Although they are completely different, they are similar in form)

Here, I use the definition of quasi-inverse from https://planetmath.org/QuasiinverseOfAFunction

Let f:X→Y be a function from sets X to Y. A quasi-inverse g of f is a function g such that

1. 
   

   g:Z→X where ran⁡(f)⊆Z⊆Y, and

   
   


2. 
   

   f∘g∘f=f, where ∘ denotes functional composition operation.

   
   
   

Note that ran⁡(f) is the range of f.

In order to understand this formula intuitively, I drew the following diagram

This formula seems to tell us that

A function $g$ is a quasi-inverse of a function $f$, if the restriction of $g$ to $ran(f)$ is the right-inverse of $f$, i.e.

$f ∘ g ∘ j_{ran(f)} = j_{ran(f)}$

Note: $j_{S}$ denote identity function on $S$.

My first question is, is this conclusion correct? i.e.

$f ∘ g ∘ j_{ran(f)} = j_{ran(f)} \Leftrightarrow f∘g∘f=f$

If this conclusion is correct, how to prove it?

It is easy to prove $\Rightarrow$, but how to prove the opposite?

If this conclusion is wrong, anyone can give me an example which satisfies $f∘g∘f=f$ but not satisfies $f ∘ g ∘ j_{ran(f)} = j_{ran(f)}$?

I may have missed some key things...

The second question is, if I have $f∘g∘f=f$ and $g∘f∘g=g$, is there any interesting conclusion? e.g. it can be concluded that f and g are bijection?

Very thanks.

PS: The reference of https://planetmath.org/QuasiinverseOfAFunction mentioned a book "Probabilistic Metric Spaces". In this book, the author mentioned another definition of quasi-inverse, which is stronger than the two quasi-inverses here, but it is another topic.