Definition 1. A category is called pointed if it has a zero object, i.e. if it has an initial object and a terminal object and they are isomorphic.
Definition 2. For $\mathcal{C}$ a category with terminal object $\ast$, the coslice category $\mathcal{C}^{\ast/}$ is the corresponding category of pointed objects: its
objects... are morphisms in $\mathcal{C}$ of the form $\ast \overset{x}{\to} X$ (hence an object $X$ equipped with a choice of point; i.e. pointed object).
morphisms... in $\mathcal{C}$ which preserve the chosen points.
Suppose that $\mathcal{C}$ is pointed.
$\mathcal{C}$ is equivalent to its category of pointed objects $\mathcal{C}^{\ast/}$ ?
What is the relationship between these two definitions?