Every hash function is NOT injective. Hashes map a large domain to a significantly smaller codomain. By the pigeon-hole principle, such a function can't be injective, because there would be items in the domain that'll map to the same item in the codomain.

Is a hash function Surjective?

Hash functions are unidirectional, a surjective function that does not allow reverse, so it is easy to calculate the hash value from any input of arbitrary length. In contrast, it is harder to return to the previous value of the message from a given hash value.

Is SHA256 injective?

SHA-256 is almost certainly not injective on 256-bit inputs, so it is almost certainly not a bijection or a surjection onto 256-bit outputs either.