So I am new to category theory and I am doing some exercises in this book.
Show that a morphism $f:x\rightarrow y$ is a split epimorphism in a category $\mathcal{C}$ if and only if for all $c\in\mathcal{C}$, post composition $f_*:\mathcal C(c,x)\rightarrow\mathcal C(c,y)$ defines a surjective function.
Here is my attempt:
"$\Rightarrow$": Suppose $f$ is a split epimorphism and $s:y\rightarrow x$ is a right inverse of $f$, that is $fs=1_y$ and let $g:c\rightarrow y$ be a morphism. Consider $t=sg:c\rightarrow x$. Then $$g=1_y g=(fs)g=f(sg)=ft$$ In other words, $t\in\mathcal C(c,x)$ is a morphism such that $ft=g$, thus $f_*$ is surjective.
"$\Leftarrow$": Consider $1_y$. By surjectivity there exists $g:y\rightarrow x$ such that $$1_y=(fg)1_y$$ This is true if and only if $g$ is a right inverse of $f$, thus $f$ is a split epimorphism.
I am unsure whether this is correct because the definition of epimorphism contains something like "if $hf=kf$ then $h=k$", but I did not use that here. I know that I need a right inverse, but is this enough?