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I am thinking about a proof of the following:

Suppose a map $f: A \to B$ has a retraction. Then for any set $T$ and for any pair of maps $x_1 : T \to A$, $x_2 : T \to A$ from any set $T$ to $A$ $$ \textrm{if } f \circ x_1 = f \circ x_2 \textrm{ then } x_1 = x_2. $$ The proof uses the diagram from the picture and I am wondering in what way the diagram shows anything? I understand the algebraic manipulations, but where does it follows from the diagram alone?