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The goal of this paper is to explain when basechange theorems for sheaves ofspaces imply basechange for sheaves with coefficients in other presentable$\infty$-categories. We accomplish this by analyzing when the tensor product ofpresentable $\infty$-categories preserves left adjointable squares. As a sampleresult, we show that the Proper Basechange Theorem in topology holds withcoefficients in any presentable $\infty$-category which is compactly generatedor stable. We also prove results about the interaction between tensor productsof presentable $\infty$-categories and various categorical constructions thatare of independent interest.

From nonabelian basechange to basechange with coefficients