Schreiber ana-inverse

Let $C$ be a category with weak equivalences and acyclic fibrations?, so that it admits a notion of ana-morphism?s.

An ana-inverse to a weak equivalence

$$A \stackrel{\simeq}{\to} B$$

is a span

$$\array{ \hat B &\to^{f}& A \\ \downarrow^{\simeq_f} \\ B }$$

such that

$$\array{ & \hat B \\ & {}^f\swarrow \downarrow^{\simeq_f} \\ A &\stackrel{\simeq}{\to}& B } \,.$$

We say a category with weak equivalences and acyclic fibrations? has ana-inverses if for every weak equivalence there exists such an ana-inverse.