  In this paper we give three functors $\mathfrak{P}$, $[\cdot]_K$ and
$\mathfrak{F}$ on the category of C$^\ast$-algebras. The functor $\mathfrak{P}$
assigns to each C$^\ast$-algebra $\mathcal{A}$ a pre-C$^\ast$-algebra
$\mathfrak{P}(\mathcal{A})$ with completion $[\mathcal{A}]_K$. The functor
$[\cdot]_K$ assigns to each C$^\ast$-algebra $\mathcal{A}$ the Cauchy extension
$[\mathcal{A}]_K$ of $\mathcal{A}$ by a non-unital C$^\ast$-algebra
$\mathfrak{F}(\mathcal{A})$. Some properties of these functors are also given.
In particular, we show that the functors $[\cdot]_K$ and $\mathfrak{F}$ are
exact and the functor $\mathfrak{P}$ is normal exact.
