**Quillen's Theorem A** is formulated as follows:


Let $F:X\to Y$ be a functor between small categories. Suppose for each $y\in Y$ the category
$F/y$ is contractible. Then $F$ induces a weak equivalence between the nerves $N(X)\to N(Y)$.


I am not a topologist, but it seems can prove the following statement, which I believe is much more powerful:


**Relative Theorem A.**
Let $F:X\to Y$ and $G:Y\to Z$ be functors between small categories. Suppose for each $z\in Z$ the induced functor $(G\circ F)/z\to G/z$ induces a weak equivalence on the nerves. Then $F$ induces a weak equivalence on the nerves $N(X)\to N(Y)$.


When $G$ is the identity functor we recover the usual Theorem A.
This seems to be so basic that it must be in textbooks. Is it well-known?



Relative version of Quillen's theorem A
The condition that appears in the assumption of what you call "Relative Theorem A" was introduced by Grothendieck in *Pursuing Stacks*. It is a part of the definition of a [basic localizer](https://ncatlab.org/nlab/show/basic+localizer), i.e. a class of functors between small categories that behaves like the class of weak homotopy equivalences.


Grothendieck posed a conjecture that weak homotopy equivalences form the smallest basic localizer which was eventually proven by Cisinski [here](http://www.math.uiuc.edu/K-theory/0601/). In particular, Théorème 2.1.13 shows (already a well-known) fact that weak homotopy equivalences indeed form a basic localizer, i.e. that the "Relative Theorem A" holds.


