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

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.