How many homomorphisms $f\colon Z_{99} \to Z_{99}$ are there?

Question

I got this question earlier:

How many homomorphisms $f\colon Z_{99} \to Z_{99}$ are there?

Which I answered with 100: First you have the identity homomorphism, and all it's shifts ($x\to x+1$, $x\to x+2$, ..., $x\to x+98$) for a total of $99$ homomorphisms.

Then all that's left is to add the trivial homomorphism: $f(x)=0$. Which brings the total up to $100$.

The answer just said $99$ though, so where am I going wrong here?

Answer 1

There are $99$, each one determined by where you send a generator, say $h(1)\in\Bbb Z_{99}$.

(If $h(1)=0$ you get the trivial one.)