Implication of retractions existing for a map

Question

Definition

1. Let $f$ be a map $A \to B$. Retraction for map $f$ is a map $r$ such that $r \circ f = I_{A}$.
2. Let $f$ be a map $A \to B$, $h$ a map from $A \to C$. Then the problem of finding a map $g$ $B \to C$ such that $g \circ f = h$ is called a determination problem.
3. Let $f$ be map $A \to B$, $h$ a map $C \to B$. Then the problem of finding a map $g$ $C \to A$ such that $f \circ g = h$ is called a choice problem.

Background

I have started reading category theory by William Lawvere. There is a theorem in it which states if map $f$ $A \to B$ has a retraction $r$, then we can always find a solution to the determination problem, ie. we can find the concerned map $g$ ( as shown below ).

My question

The theorem does not tell if there exists a retraction $r$ for map $A \to B$ then the choice problem also has a solution. That is we can find the concerned map $g$ ( as shown below ). Am I missing something or am I wrong about the result the theorem does not tell. Is there a reason why the author links solution of retraction problem to a determination problem ( and not to a choice problem ) ?

Answer 1

In your question you haven't solved the choice problem: you have no grounds to conclude $$f\gamma h = h$$ In fact, by considering the particular case of $h = 1_B$, you have $$f \gamma 1_B = 1_B$$ if and only if $\gamma = f^{-1}$.

Answer 2

Suppose $f: A→B$ and f has a retraction $r$. And further suppose that $h:C→B$ is given. Problem is to find $g$ such that $f∘g=h$ (1) Now, take $$g=r∘h$$. Substituting in (1), $$f∘(r∘h)=h$$ Solving left hand side, Using associative identity, $$(f∘r)∘h=1_B∘h$$

This is the step where you assumed that equality always holds, and you saw that $1_B∘h=h$. But recognize in the definition of retraction, that $r$ is retraction of $f$ if $r∘f=1_A$. Counter Example: Suppose that if $r$ is not inverse of $f$, then $f∘r=1_B$ is not true and $r$ is still a retraction. (Because if $f∘r=1_B$ is true then r is inverse of f, which will contradict our assumption).