In Awodey's Category Theory (2nd edition), page 229, I read:



> 
> the category of elements $J$ of a representable $yC$ has a terminal
>  object, namely the element $1\_C \in Hom\_{\mathbf{C}}(C,C)$
> 
> 
> 


However, I don't see how this is possible without assuming that all elements of $J$ are split epis. 


Indeed, the quoted passage means that:
$\forall D \in \mathbf{C}, \forall x \in Hom\_{\mathbf{C}}(D,C)$, there is an (unique) arrow from $x$ to $1\_C$. Am I correct?


An arrow from $x \in Hom(D,C)$ to $1\_C \in Hom(C,C)$ would be of the form $f^\*$ with $f \in Hom(C,D)$.
We would have: $f^\*(x)=x \circ f$. So, $x \circ f = 1\_C$, so that $x$ is a split epi.


Could somebody help me find what is wrong with my reasoning, or show how to correct this passage from Awodey?



Terminal object of category of elements of a representable functor
My initial answer was based on the same misconception, which I discovered was a misconception by digging farther back to p.196 in Awodey. An arrow $g:\langle f, Hom(A,C)\rangle \to \langle f', Hom(A',C)\rangle$ doesn't come from a $\mathbf{C}$-arrow $A'\to A$, but actually one $g:A\to A'$ after all, with $Hom(g,C)(f')=f$. The composition rule you describe is that of $(1\downarrow Hom(-,C))$, where the category of elements of $yC$ is rather the dual of that comma category (I think).


