[hcrypt]

>FHE extends HE so that you can perform arbitrary computation

FHE doesn't extend HE in the sense that it adds operation types other than addition and multiplication but it allows unlimited chaining of the two operations. This in turn enables arbitrary operations because you can express any function (i. e. any chain of atomic operations) by means of additions mod 2 and multiplications mod 2 (which is equivalent to XOR and AND) when you turn it into a boolean circuit representation. The contribution of FHE is a way to a) clean operands from noise or to b) do not add noise to the operands in the first place because noise is the limiting factor for circuit depth in HE, where "depth" means the number of possible subsequent multiplications.