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In their moduli of $p$-divisible groups paper, Scholze and Weinstein show that a certain diagram of perfectoid spaces is Cartesian. In this paper, we generalize their result. This generalization will be used in a forthcoming paper of ours to compute certain non-trivial $\ell$-adic \'etale cohomology classes appearing in the the generic fiber of Lubin-Tate and Rapoport-Zink towers. We also study the behavior of the vector bundle functor on the fundamental curve in $p$-adic Hodge theory, defined by Fargues-Fontaine, under multilinear morphisms.
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