A name for morphisms with the same source object

Question

Let $G$ be a groupoid (or, more generally, a category). Given an object $o \in \mathrm{Obj}(G)$, is there a name for the set of morphisms having $o$ as their source object?

The motivation is the following. Given a (nice) topological space $X$, consider its fundamental groupoid $\Pi_1(X)$. Fixing a basepoint $o \in X$, the points of the universal cover $\widetilde{X}$ of $X$ can be identified with the morphisms of $\Pi_1(X)$ having $o$ as their source object. Moreover, the fundamental group $\pi_1(X,o)$ naturally acts on $\widetilde{X}$ through its left-multiplication on the previous morphisms. This suggests that such sets of morphisms could deserve their own name.

Answer 1

In his notes on algebraic topology in the section on covering spaces, Peter May seems to refer to this as a star.

Answer 2

I don't know if there's a name for the set of morphisms out of some object $X$ in a category $\mathscr{C}$, but very often you want to consider the category of morphisms out of $X$. The objects in this category consist of an object $Y$ together with a morphism $f\colon X\to Y$, and a morphism from $f\colon X\to Y$ to $g\colon X\to Z$ is a morphism $k\colon Y\to Z$ such that $X\xrightarrow{f}Y\xrightarrow{k}Z = X\xrightarrow{k}Z$. This is called the coslice category under $X$.

Alternatively, you could consider the functor $\mathscr{C}(X,{-})$, which you can think of as taking all the morphisms out of $X$ and organising them into a functor $\mathscr{C}\to \mathbf{Set}$. If you take the category of elements of this functor you get the coslice category under $X$. Such functors are very important as they appear in the Yoneda lemma.