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Yes, this is correct. It's a really simple principle, and I think an explanation can help you understand :)

Suppose you have 3 holes, and 4 pigeons, and you stuff the pigeons into holes. There must be 1 hole with at least 2 pigeons, right?

The same is true with hash functions. If you're hashing data down to a fixed length, like say 256-bits with sha-256, and the data is longer than 256 bits, there must be a collision somewhere.

Wouldn't this imply that all hash functions (other than one-to-one mappings) must have collisions?

Yes, they do. Finding them is the hard part.

Yes; by definition using something with X possible values to represent something with Y possible values will always have collisions if X < Y.