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?