**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?

Le localisateur fondemental minimal]. $\endgroup$ – Zhen Lin Mar 31 '16 at 22:30