Is there a deeper (categorical) reason for this?
On the one hand a group homomorphism $\phi:(G,\cdot)\to (H,\star)$ preserves 'results of operations' as well as the identity element and inverse elements, but satisfies only one equation: $$\forall g_1,g_2\in G:\phi(g_2\cdot g_1)=\phi(g_2)\star\phi(g_1)$$
while if $\phi$ was a monoid homomorphism instead, it would only preserve the first two things, but need to satisfy the additional equation $$\phi(e_G)=e_H$$
I know how to prove algebraically that a group homomorphism preserves all the mentioned structure, that's not the question. I do not understand why it preserves more structure than the monoid homomorphism, while at the same time having less 'algebraic conditions'.
EDIT: I think what really gives me trouble is that going 'in the natural order' from semigroups to monoids to groups, one starts with one equation for semigroup-homomorphisms, then adds an additional equation for monoid-homomorphisms, and then for group-homomorphisms one goes back to one equation. This seems strange to me.
Best Answer
I'm not sure about a "categorical" reason, but generally if you restrict to nice subclass of objects, you have less "bad behavior" and so you're theorems/definitions need less restrictions/hypotheses.
As an extreme, what if we just considered the category of "Trivial Groups"? Then every function is a homomorphism! So we don't even need to specify that maps are operation preserving.
An analogy: In general, functions can be 1-to-1 or not, onto or not. But what if we restrict our attention to sets of size $15$? Then a function from a set of size $15$ to a set of size $15$ is 1-to-1 iff it's onto. Thus in my world of size $15$, I can define bijections to be 1-to-1 functions (I get onto for free). The definition of "bijection" is simplified merely because I've moved into a very restrictive world where the phenomena of 1-to-1 and onto are equivalent.