Can anybody give me an independent proof for the universal property of tensor product for the direct limit $B$ in the picture without using the argument that the tensor product of algebras over a commutative ring is the coproduct.
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What, for you, is the universal property of the tensor product of an infinite family of $A$-algebras? – Alex Kruckman Feb 01 '20 at 05:25
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4I've voted to close as a duplicate of your other question Direct limit of arbitrary family of tensor products of A-algebras - but I'll write an answer there. – Alex Kruckman Feb 01 '20 at 05:32