# 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?