cannot recall seeing a proof of associativity, of anything, that didn't make my eyes glaze over
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the tensor product of vector spaces is associative (up to isomorphism) by the yoneda lemma, because (V_1 \otimes V_2) \otimes V_3 and V_1 \otimes (V_2 \otimes V_3) have the same universal property, namely, they both represent trilinear maps out of V_1 x V_2 x V_3