(I) 1) Because \(f:A\longrightarrow B\) is a section, exists \(g:B\longrightarrow A\) such that \(g\circ f=id_A\). $$\xymatrix{ A \ar[r]^f \ar[dr]_{id_A} & B \ar[d]^g \\ & A }$$ If for \(g_1,g_2:C\longrightarrow A\) holds \(f\circ g_1=f\circ g_2\), $$\begin{xy} (0,0)*+{C}="a", (20,0)*+{A}="b", (40,0)*+{B}="c" \ar @{>}^{g_1} "a";"b" < 2pt> \ar @{>}_{g_2} "a";"b" <2pt> \ar @{>}^f "b";"c" \end{xy}$$ then \begin{align*} g\circ (f\circ g_1)=g\circ (f\circ g_2)\quad&\Rightarrow\quad (g\circ f)\circ g_1=(g\circ f)\circ g_2\\ &\Rightarrow\quad id_A\circ g_1=id_A\circ g_2\\ &\Rightarrow\quad g_1=g_2\,. \end{align*} So, \(f\) is monic.
2) Because \(f:A\longrightarrow B\) is a retraction, exists \(g:B\longrightarrow A\) such that \( f\circ g=id_B\). $$\xymatrix{ A \ar[r]^{f} & B \\ B \ar[u]^g \ar[ur]_{id_B} & }$$ If for \(g_1,g_2:B\longrightarrow C\) holds \( g_1\circ f=g_2\circ f\), $$\begin{xy} (0,0)*+{A}="c", (20,0)*+{B}="b", (40,0)*+{ C}="a" \ar @{>}^f "c";"b" \ar @{>}^{g_1} "b";"a" < 2pt> \ar @{>}_{g_2} "b";"a" <2pt> \end{xy}$$ then \begin{align*} (g_1\circ f)\circ g =(g_2\circ f)\circ g \quad&\Rightarrow\quad g_1\circ( f\circ g) =g_2\circ( f\circ g)\\ &\Rightarrow\quad g_1\circ id_B=g_2\circ id_B\\ &\Rightarrow\quad g_1=g_2\,. \end{align*} So, \(f\) is epic.
_________________ Grigorios Kostakos
