Let $\operatorname{C}$ be a category and $c\in \operatorname{C}$ an object. Consider the coslice (sometimes called slice under) category ${\operatorname c}/{\operatorname C}$. My question is whether ${\operatorname c}/{\operatorname C}$ is in general a *full* subcategory of $\operatorname{C}$?
I am particularly interested in the case of commutative $\operatorname{R}$-algebras as the slice category ${\operatorname R}/{\operatorname{CRing}}$, and thus whether the category $\operatorname{R-Alg}$ of commutative $\operatorname R$-algebras is a full sub category of $\operatorname{CRing}$, the category of commutative rings.



Is a coslice (slice under) category a full subcategory of it ambient category?
For a counterexample, consider $R = \mathbb{C}$. The automorphism $i \colon \mathbb{C} \to \mathbb{C}$ given by complex conjugation is certainly a ring homomorphism, but it isn't $\mathbb{C}$-linear.


Edit: Another example would be the slice $p / \text{Set}$, where $p$ is a one element set. This category is the category of pointed sets, and not all maps between two sets will preserve the chosen point.


