Going through Emily Riehl's Category Theory in Context and something keeps tripping me up.
The notation for slice categories, which reminds me of factor group notation in group theory, indicates to me that the morphisms in $c/C$ are in fact morphisms inherited from $C$. As a more specific example/question:
If G is a fixed group in Group and we have homomorphisms $f : G \xrightarrow{} H$ and $g : G \xrightarrow{} K$ in Group, and then we have G/Group where morphisms are $h : H \xrightarrow{} K$, is $h$ also a group homomorphism or is it a differently structured morphism?
I don't think it's explicitly said in the book whether the morphisms are inherited from the original category.
A morphism in this category should be a morphism $h: H \to K$ in the original category fitting into a commutative triangle with the given morphisms $G \to H$ and $G \to K$.
(i.e., yes, we inherit the morphisms, but we don't inherit all of them!)