Identity function

I've been struggling for a while now to understand the meaning of the concept of Identity Function in category theory. It figures in the basic axioms of the theory but there's barely an explanation of why or how to use it. The explanations I've seen so far are purely formal. The function is simply there: f(a) = a.

This morning I finally came to the conclusion that Identity Function is something (a map, a transformation, a relationship) that doesn't make a difference. It sounds obvious; nevertheless, the idea to define the identity of an object through something that leaves it indifferent is completely non–intuitive to me. It stands in contrast with the classic approach to defining objects through their essences, James Joyce's "Horsiness is the whatness of allhorse" being a somewhat mock example of it.

upd: from this perspective, picking an abstraction level is critical because it changes the observables, thus changing the identity function.

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"Quizás,Quizás,Quizás - Andrea Bocelli - Jennifer Lopez" (Watch on YouTube)

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Now, that I understand identity functions (wink-wink), I can say that voter turnout numbers in Russian elections don't make a difference. Moreover, the whole point of Russian elections is to render votes of all citizens useless, whether they have actually voted or not. That was the case during the Soviet Union times and that is the case now. Russia's identity function is that all imported western democratic institutions make no difference whatsoever.

In today's US, a candidate for the identity function would be an attempt to amend the Constitution - makes no difference at all.