Consider the map $\varphi:\{0,1,2,3\}\to\{0,1,2,3\}$ given by $\varphi(x)=x$. If $\{0,1,2,3\}$ is endowed with the structure of $\mathbb Z/4\mathbb Z$, then $\varphi$ may be considered to be a group homomorphism between $\mathbb Z/4\mathbb Z$ and $\mathbb Z/4\mathbb Z$.
Now consider the map $\psi:\{0,1,2,3\}\to\{0,1,2,3\}$ given by $\psi(x)=x$, and suppose $\{0,1,2,3\}$ is endowed with the structure of the Klein-four group $K_4$. Then, $\psi$ may be consider a group homomorphism between $K_4$ and $K_4$.
If we define a function as an ordered triple $(A,B,f)$ satisfying certain conditions, then $\varphi$ and $\psi$ are clearly equal. However, to me it seems natural to consider them to be different homomorphisms, since the groups are different – it's just that their underlying sets happen to be equal. In other words, I am inclined to formally define a group homomorphism as an ordered triple $((G,\cdot),(H,\star),f)$, where $(G,\cdot)$ and $(H,\star)$ are both groups, and $f$ is the graph of a function $\varphi:G\to H$ satisfying $\varphi(a)\star\varphi(b)=\varphi(a\cdot b)$.
Although this objection might seem unnecessarily pedantic, there seem to be genuine issues with defining homomorphisms as functions. If we consider $\varphi$ and $\psi$ to be the same homomorphism, then in the category of groups, what is the source of $\varphi$? This should be a group with the underlying set $\{0,1,2,3\}$, but it seems impossible to "know" which group it should be.
My question is: should the "correct" definition of a group homomorphism, and more generally a homomorphism between two algebraic structures, be a function satisfying certain constraints? Or is my technical objection correct, but in practice we are too lazy to care about such things?
There is one way to define a category that should resolve your concern : A category is the data of a class Ob(C) of objects and a class Mor(C) of morphisms, with two maps $s,t : Mor(C) \to Ob(C)$, namely "set and target". (There are obviously also axioms for composition of morphisms, identity, etc). Then $Hom(x,y)$ can be defined as the subset (subclass if you are picky about set-theoretic problems) of elements $f \in Mor(C)$ such that $s(f) = x$ and $t(f) = y$.
Therefore, in the category of groups, the source of $\phi$ is not only the set $G$ but the group $(G,\cdot)$. So concretely, $\phi$ and $\psi$ considered as group morphisms, are different maps simply because their sources are not equal. If you are concerned that $\phi$ and $\psi$, as maps (so as binary relations on $G^3$, etc) are identical, we can choose to define a group morphism as the set-theoretic data of the map plus the data of group law.