By a directed multigraph, I mean a tuple $G= (V,E,i,t)$, where $V$ is a set of "vertices," $E$ is a set of "edges," and $i:E\to V$ and $t:E\to V$ are functions giving the "initial" and "terminal" vertices of edges.
A map $f:G\to H$ between directed multigraphs thus takes directed edges to directed edges, preserving orientation. We'll say $f$ is an outmersion if, whenever $e_1$ and $e_2$ are distinct edges with $i(e_1)=i(e_2)$, we have $f(e_1)\ne f(e_2)$. In other words, $f$ is $1$-$1$ on the set of edges emanating from $v$ for each vertex $v$ of $G$.
So $f$ is locally injective on outgoing edges, but not necessarily on incoming edges. It is thus an "outward-immersion" if we think of germs of edges as constituting the tangent space at each vertex.
Are there references to this notion, and is there an accepted term for an outmersion?