How do you notate the edges connected to a node in a graph?

So given a directed multigraph $G=(V,E)$ I am notating the nodes of any edge like so $(e_s,e_t) = e \in E$ where $e_s$ is the source and $e_t$ is the target.

However I am unsure how to notate the edges of a node that finish or start there. So far I have written $v_\text{in} = \{e_1,...,e_k\} \subseteq E$ where $v_\text{in}$ is all edges finishing at node $v$. However it's not explicit that the edges in $v_\text{in}$ actually finish at node $v$.

Is there a standard convention or more accepted way of expressing this?

Also I'm in the computer sciences so it would be great to know if I have made any mistakes in expressing the maths involved in this question.

Edit: Might have answered my own question, can I just write $v_\text{in} = \{e \mid e_t = v \}$?

Defining $v_{\text{in}} = \{e \mid e_t = v\}$ seems reasonable. (Even explaining in words "$v_{\text{in}}$ is the set of edges that end at $v$" would be fine.) It's not notation that I've seen people define before, but that's not a problem if you give a definition. My only doubt about it is that it's unusual for lowercase letters to denote sets.
It's fairly common to define $E(S,T) = \{e \mid e_s \in S \text{ and } e_t \in T\}$: the set of edges from $S$ to $T$. If you do so, you could talk about $E(V, \{v\})$ for the set of edges that can start anywhere and end in $v$. This is a bit more cumbersome, but it's more flexible and will be familiar to some people.
Relatedly, $N^+(v)$ and $N^-(v)$ denote the set of "out-neighbors" and "in-neighbors" of $v$, respectively: $N^+(v)$ is the set of vertices that can be reached from $v$ by an edge, and $N^-(v)$ is the set of vertices which can reach $v$ by an edge. This isn't quite what you want. But if you, analogously, define $I^+(v)$ and $I^-(v)$ for the edges out of and into $v$, nobody will be too surprised. (I'm not convinced $I$ (for "incident") is the best letter to use here, but it's not the worst.)
I would suggest using the notation $e_{ijk}$ to denote the $k^{th}$ edge that starts from vertex $v_i$ and ends at $v_j$. I come from a physics background and this notation is very popularly used in the network modeling of complex systems.