  The main objective of the present paper is to present a version of the
Tannaka-Krein type reconstruction Theorems: If $F:B\to C$ is an exact faithful
monoidal functor of tensor categories, one would like to realize $B$ as
category of representations of a braided Hopf algebra $H(F)$ in $C$. We prove
that this is the case iff $B$ has the additional structure of a monoidal
$C$-module category compatible with $F$, which equivalently means that $F$
admits a monoidal section. For Hopf algebras, this reduces to a version of the
Radford projection theorem. The Hopf algebra is constructed through the
relative coend for module categories. We expect this basic result to have a
wide range of applications, in particular in the absence of fibre functors, and
we give some applications. One particular motivation was the logarithmic
Kazhdan-Lusztig conjecture.
