A bijection is a function between two sets where each element in the preimage has exactly one element in the image and visa versa. This means all elements in the preimage and all objects in the image have a unique pair which the function transforms from preimage set to image set. A bijective mapping is both surjective and injective.
definition: An isomorphism is a mapping between two mathematical groups which maintains sets and relations of the elements of the groups. If two mathematical objects have an isomorphism this means they are effectively the same objects after all the elements are renamed (through the isomorph mapping). More formally an isomorphism is a bijective morphism.