Notation.
Using a downarrow to denote comma categories (e.g.), we can write $\mathbf{Set} \downarrow X$ for the slice category of a set $X$ and $X \downarrow \mathbf{Set}$ for the coslice category.
What I know.
I'm pretty sure that slice categories (in the world of sets) can be described as follows: $\mathbf{Set} \downarrow X$ is initial small-coproduct category equipped with a distinguished object for each element of $X$.
What I'm wondering about.
I'm wondering there's a similar description of the coslice categories (also in the world of sets). It seems to me that $X \downarrow \mathbf{Set}$ can be described as the initial small-coproduct category equipped with a distinguished object $1$ and a distinguished morphism $1 \rightarrow 0$ for each element of $X$, where $0$ is the initial object. The intended meaning of $1$ is that it ends up being $$(\{*\} \sqcup X, \eta_2 : X \rightarrow \{*\} \sqcup X).$$
Question. Is this correct? If not, what is the initial thing described in the previous paragraph?
One though I had was that maybe we should only be considering injective coslices. That is, our objects are injective functions from $X$ to another set. Morphisms are arbitrary functions making the relevant triangle commute.