The use of the equals sign is an abuse of notation; those two groups are not really identical but there is a way of identifying them that is natural. It can be seen to be natural on the basis, firstly, that these are two alternative descriptions of the bilinear mappings from X × A to Y. That is, however, something particular to the case of tensor product. In category theory the 'naturality' of the bijection is subsumed in the concept of a natural isomorphism.
The use of the equals sign is an abuse of notation; those two groups are not really identical but there is a way of identifying them that is natural. It can be seen to be natural on the basis, firstly, that these are two alternative descriptions of the bilinear mappings from X × A to Y. That is, however, something particular to the case of tensor product. In category theory the 'naturality' of the bijection is subsumed in the concept of a natural isomorphism.