Let's start from directed graphs. There exists notion of structure preserving map $\sigma$ between graphs $\mathbb{G}$ and $\mathbb{G'}$ defined as follow: $$\forall arrow \in Arrows(\mathbb{G}): \sigma(arrow) \in Arrows(\mathbb{G'})$$
Note that it could be expanded in much more clear definition, yet the essence is the same: homomorphism is not allowed to tear apart arrows with respect to their source and destionation vertices.
- Consider $\mathbb{H}$ - directed graph with single vertex $*$ and single arrow $* \rightarrow *$. For arbitrary directed graph $K$ there exists absolutely valid homomorphism from $\mathbb{K}$ to $\mathbb{H}$ that sends every vertex of $\mathbb{K}$ to the $*$ and every arrow from $\mathbb{K}$ to the $* \rightarrow *$. Even though it indeed preserves structure to certain extend, it actually forgets everything about $\mathbb{K}$ except that it has "at least one node and at least zero arrows". So in some sense (which feels more important to me), structure is lost.
- Consider graph $\mathbb{C}$ with two vertices and single arrow $1 \rightarrow 2$. Let graph $\mathbb{C+}$ be an extension of the $\mathbb{C}$ with one more arrow $2 \rightarrow 1$. Then one could build a dead simple homomorphism $\delta: \mathbb{C} \mapsto \mathbb{C+}$ by taking $1 \in \mathbb{C}$ to $1 \in \mathbb{C+}$ and $2 \in \mathbb{C}$ to $2 \in \mathbb{C+}$ and $1 \rightarrow 2 \in \mathbb{C}$ to $1 \rightarrow 2 \in \mathbb{C+}$, thus introducing $2 = \delta(2 \in \mathbb{C}) \rightarrow 1 = \delta(1 \in \mathbb{C})$ which did not exist in $\mathbb{C}$ at all. So "structure preserving map" was allowed to grow structure silently, "on the fly".
I am not a mathematician and wondering how such definition of homomorphism is justified: what's the way of thinking and intuition behing it?
P.S. Functor, seen as structure preserving map between categories, allows both kind-of "anomalities" I've described, so I don't bother to translate them into a category theory formalism.