nLab composable pair

A composable pair of morphisms in a given category $C$ consists of objects $X,Y,Z$ of $C$ and a pair of morphisms $f\colon X \to Y$ and $g\colon Y \to Z$. The composite of this composable pair is the morphism $g \circ f\colon X \to Z$.

A composable pair in $C$ is precisely a 2-simplex in the nerve of $C$.

Sometimes one defines a composable pair to be a literal pair $(f,g)$ such that the target of $f$ is equal to the source of $g$, but this violates the principle of equivalence