I would like to know the standard terminology for the following two notions.
Notion 1: $E_1\to B$ and $E_2\to B$ are fibrations over the same base space, and $f\colon E_1\to E_2$ is a map making the evident triangle commute.
Notion 2: $E_1\to B_1$ and $E_2\to B_2$ are fibrations over possibly different base spaces, and $f\colon E_1\to E_2$ and $\phi\colon B_1\to B_2$ are maps making the evident square commute.
Of course, notion 1 is the special case of notion 2 where $\phi$ is an identity map.
Some phrases I can think of that might be used to describe either of the two notions are:
- map of fibrations
- parametrized map
- fiberwise map
- fiber-preserving map
Is there a standard convention in algebraic topology regarding which phrase refers to which notion?