Consider two morphisms: $f : X \to Y$ and $g : Y \to Z$ , and their composition: $g \circ f : X \to Z$.
What is the name given to the role of $Y$ with respect to $g \circ f$? Is there a naming convention to distinguish between binary composition and the intermediary terms from composition operations of higher arity?
Normally, if we have a morphism $f\colon X\rightarrow Z$ such that there exist morphisms $l\colon X\rightarrow Y$ and $r\colon Y\rightarrow Z$, such that $f=r\circ l$ we say that the morphism $f$ factors through $Y$ via $l$ and $r$. Is this the sort of terminology you're looking for?