  Viewing group cohomology as a so-called cohomological functor, G. Mislin has
generalised Tate cohomology from finite groups to all discrete groups by
defining a completion for cohomological functors in [24]. For any cohomological
functor $T^{\bullet}: \mathcal{C} \rightarrow \mathcal{D}$ we have constructed
its Mislin completion $\widehat{T}^{\bullet}: \mathcal{C} \rightarrow
\mathcal{D}$ in [15] under mild assumptions on the abelian categories
$\mathcal{C}$ and $\mathcal{D}$ which generalises Tate cohomology to all $T1$
topological groups. In this paper we investigate the properties of Mislin
completions. As their main feature, Mislin completions of Ext-functors detect
finite projective dimension of objects in the domain category. We establish a
version of dimension shifting, an Eckmann-Shapiro result as well as cohomology
products such as external products, cup products and Yoneda products.
