In homotopy theory, the *mapping cone* of a continuous map $f\colon X \to Y$ is the homotopy pushout over the following span:

$$ \require{AMScd} \begin{CD} X @>{f}>> Y\\ @VVV \\ \{*\} \end{CD} $$

I.e., it is universal among all squares of the form $$ \begin{CD} X @>{f}>> Y\\ @VVV @VVV\\ \{*\}@>>> Z \end{CD} $$ where the square commutes up to homotopy.

But what is a good name for such an object $Z$? Normally, I would call it a **cocone**, but I would rather not use the word *cone* to mean two different things.

*Square* and *cospan* are possibilities, but they seem a bit too general: I want to refer specifically to cocones for the first diagram.

Is there a good alternative word?