I am trying to teach myself some category theory so I apologize if this question is extremely basic. The question is:
Show that a morphism $f: x \rightarrow y$ is a split epimorphism in a category C if and only if for all $c \in$ C, post-composition $f_* :$ C$(c,x) \rightarrow $ C$(c,y)$ defines a surjective function.
Since $f$ is a split epimorphism, there is a morphism $g : y \rightarrow x$ such that $fg = 1_y$. However, I am not understanding how this helps prove surjectivity. I'm not entirely sure how the post-composition is to be defined here; I imagine it is something along the lines of a morphism in C$(c,x)$ post-composed with $f$ so we get a morphism in C$(c,y)$. And then I should try to show that for every morphism in C$(c,y)$, such a composition is possible and so $f_* :$ C$(c,x) \rightarrow $ C$(c,y)$ is surjective?
I am also not sure about how to prove the other direction.
Any help or clarifications would be appreciated! Thank you!
Post-composition by $f:x\to y$ is $f_*:\mathcal C(c,x)\to\mathcal C(c,y)$ defined by $f_*(g)=fg$.
For the foward direction: For $h:c\to y$, you want an arrow $(-):c\to x$ s.t. $f(-)=h$. Can you think of such an arrow?
Other direction: You want a right inverse $g$ to $f$ i.e. $fg=1_y$. How would you get $1_y$ to appear?