  We show that Kazhdan and Lusztig's category $KL^k(\mathfrak{sl}_2)$ of
modules for the affine Lie algebra $\widehat{\mathfrak{sl}}_2$ at an admissible
level $k$, equivalently the category of finite-length grading-restricted
generalized modules for the universal affine vertex operator algebra
$V^k(\mathfrak{sl}_2)$, is a braided tensor category. Although this tensor
category is not rigid, we show that the subcategory of all rigid objects in
$KL^k(\mathfrak{sl}_2)$ is equal to the subcategory of all projective objects,
and that every simple module in $KL^k(\mathfrak{sl}_2)$ has a projective cover.
Moreover, we show that the full subcategory of projective objects in
$KL^k(\mathfrak{sl}_2)$ is monoidal equivalent to the category of tilting
modules for quantum $\mathfrak{sl}_2$ at the root of unity $\zeta=e^{\pi
i/(k+2)}$. Using this, we establish a universal property of the tensor category
$KL^k(\mathfrak{sl}_2)$, and as an application, we prove a weak Kazhdan-Lusztig
correspondence, that is, we obtain an exact essentially surjective (but not
full or faithful) tensor functor from $KL^k(\mathfrak{sl}_2)$ to the category
of finite dimensional weight modules for the quantum group associated to
$\mathfrak{sl}_2$ at the root of unity $\zeta $. We also use the universal
property to classify the categories $KL^k(\mathfrak{sl}_2)$ up to (braided)
tensor equivalence and to obtain a tensor-categorical version of quantum
Drinfeld-Sokolov reduction, that is, we construct a braided tensor functor from
$KL^k(\mathfrak{sl}_2)$ to a category of modules for the Virasoro algebra at
central charge $1-\frac{6(k+1)^2}{k+2}$.
