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In 2018, Morrison, Walker, and Wedrich defined an extension of Khovanov-Rozansky homology to invariants of smooth 4-manifolds. These invariants, called skein lasagna modules, can be computed for 2-handlebodies via a colimit of cables associated to the attaching link in the Kirby diagram. In joint work with Melissa Zhang (UC Davis), we lift this colimit of cables construction to the level of Bar-Natan's tangles and cobordisms, working instead with homotopy colimits in the Bar-Natan category. Our local techniques allow for new computations of the $N=2$ skein lasagna module for 4-manifolds whose Kirby diagrams contain a 0-framed meridional unknot component. Using these techniques, we partially answer a conjecture of Manolescu that states that the skein lasagna module of $S^{2}\times{S^{2}}$ is 0.