  Given a non-semisimple braided tensor category, with oplax tensor functors
from known braided tensor categories, we ask : How does this knowledge
characterize the tensor product and the braiding? We develop tools that address
this question. In particular we prove that the associator is fixed by the oplax
tensor functors, and we show that a distinguished role is played by the
coalgebra structure on the image of theses tensor functors.
  Our setup constrains the form of quasi bialgebras appearing in the
logarithmic Kazhdan-Lusztig conjecture and it applies in particular to the
representation categories of the triplet vertex algebras. Here the two oplax
tensor functors are determined by two free field realizations, and the
coalgebras mentioned above are the Nichols algebras of type $\mathfrak{sl}_2$.
We demonstrate in the case of $p=2$ that our setup completely determines the
braided tensor category and the realizing quasi-triangular quasi-Hopf algebra
is as anticipated in \cite{FGR2}. This proves the logarithmic Kazhdan-Lusztig
conjecture for $p=2$, while for general $p$ it only remains to establish that
our characterization provides a unique braided tensor category.
