From nonabelian basechange to basechange with coefficients

The goal of this paper is to explain when basechange theorems for sheaves of spaces imply basechange for sheaves with coefficients in other presentable ∞-categories. We accomplish this by analyzing when the tensor product of presentable ∞-categories preserves left adjointable squares. As a sample result, we show that the Proper Basechange Theorem in topology holds with coefficients in any presentable ∞-category which is compactly generated or stable. We also prove results about the interaction between tensor products of presentable ∞-categories and various categorical constructions that are of independent interest. For example, we show that tensoring with a compactly generated presentable ∞-category preserves fully faithful or conservative left exact left adjoints. We also show that tensoring with a presentable ∞-category that is compactly generated or stable preserves recollements.