Abstract algebra – How does the equalizer (category theory) characterize the equalizer of set theory?

In the Set category, the equalizer $ f: A a B $ is the (largest) set of elements $ x in A $ such that $ f (x) = g (x) $.

But in category theory, this is generalized to a map. $ E xrightarrow {e} A $ such that $ f circ e = g circ e $ and such that the UMP maintains.

But looking back at the whole example, can not it take a subset smaller than the larger one and surely that (inclusion) would be a map? $ e $. In other words, how does an equalizer ensure that $ E $ Is it in fact the "biggest" object?