Class of weak equivalences contains the isomorphisms.

Question

I need to prove that given $C$ a category with a class of weak equivalences $W$. That is, $W$ contains the identies and it satisfies the 2 out of 6 property. Show that $W$ contains the isomorphisms.

I've found this stated in numerous papers and books. I can't seem to find a proof of it.

Answer 1

Suppose $\varphi$ is an isomorphism from $A$ to $B$, then we get the following $$A \xrightarrow{\varphi} B \xrightarrow{\varphi^{-1}} A \xrightarrow{\varphi} B$$ Since $\varphi^{-1}\varphi=1_A \in W$ and $\varphi\varphi^{-1}=1_B \in W$, we get by using the $2$ out of $6$ property the conclusion.