So, when defining a category, one is careful enough to define the identity of a object $a\in\text{Ob}(\mathcal C) $ as a particular element of the hom-class, $\text{id}_{a}\in\text{Hom}(a,a)$. That does seem natural, especially when one defines a functor. After all, it wouldn't really mean anything for a functor to preserve the identity morphism if the hom-class $\text{Hom}(a,a)$ only went around the identity.
So, yeah, I'm not relutant into accepting that. But I'm having trouble visualizing how could you even define a morphism from an object to itself that does not simply converge to the definition of an identity. I do suspect that this uncapability of mine may be me restricting my understanding to small categories.
So, how can you define a morphism in $\text{Hom}(a,a)$ without just going back to the definition of an identity? Because, once again, if the distinction is not made, I don't think the identity morphism preserved by a functor can make sense. Plus, the definition of a category to that extent, would also seem a little circular.
Any help will be appreciated.
Suppose $\mathcal{C}$ is the category whose objects are sets and whose morphisms are functions between sets with the usual composition. Then given a set $S \in \operatorname{Ob}(\mathcal{C})$, an element of $\operatorname{Hom}(S, S)$ is just a function from $S$ to itself, and of course there are typically many such functions. (For example, if $S$ is a finite set of size $n$, then there are $n^n$ functions from $S$ to itself, only one of which is the identity function.)
Many examples are similar to the above. For a different sort of example, given a group $G$, we can define a category $\mathcal{C}$ as follows: there is only a single object (call it $\bullet$), and the elements of $\operatorname{Hom}(\bullet, \bullet)$ are exactly the elements of $G$, with composition given by the group law in $G$. (This lets us think of every group as a category with one object in which all morphisms are invertible.)
Hopefully these two examples are enough to give you a sense of how this is possible. In general, because categories are such abstract, general objects, it's always a good idea to ground your understanding of category theory in concrete, familiar examples.