I’ve had recent cause to consider the following construction: given a category $\newcommand{\C}{\mathbf{C}}\C$, define a semicategory $M(\C)$, whose objects are arrows of $\C$, and where a map from $f : A \to B$ to $g: C \to D$ is just a map $h : B \to C$ in $\C$: $$ A \overset{f}{\to} B \overset{h}{\dashrightarrow} C \overset{g}{\to} D$$
This forms a semicategory — i.e. it has an evident associative composition, but lacks general identity arrows.
Has anyone encountered this construction before, and if so, is there an established terminology/notation for it? I had a bit of a look in the literature, but without any idea of the terminology, it’s hard to know what to begin searching for.
