  While the basic greedy algorithm gives a semi-streaming algorithm with an
approximation guarantee of $2$ for the \emph{unweighted} matching problem, it
was only recently that Paz and Schwartzman obtained an analogous result for
weighted instances. Their approach is based on the versatile local ratio
technique and also applies to generalizations such as weighted hypergraph
matchings. However, the framework for the analysis fails for the related
problem of weighted matroid intersection and as a result the approximation
guarantee for weighted instances did not match the factor $2$ achieved by the
greedy algorithm for unweighted instances. Our main result closes this gap by
developing a semi-streaming algorithm with an approximation guarantee of
$2+\epsilon$ for \emph{weighted} matroid intersection, improving upon the
previous best guarantee of $4+\epsilon$. Our techniques also allow us to
generalize recent results by Levin and Wajc on submodular maximization subject
to matching constraints to that of matroid-intersection constraints.
  While our algorithm is an adaptation of the local ratio technique used in
previous works, the analysis deviates significantly and relies on structural
properties of matroid intersection, called kernels. Finally, we also conjecture
that our algorithm gives a $(k+\epsilon)$ approximation for the intersection of
$k$ matroids but prove that new tools are needed in the analysis as the used
structural properties fail for $k\geq 3$.
