# Need to find a functor T : Set &#x2192;<!-- → --> Set such that Alg(T) is concretely

Need to find a functor $T:$ Set $\to$ Set such that Alg(T) is concretely isomorphic to the category of commutative binary algebras.
The first idea is that the functor is likely to map object $X\in Ob\left($ to the $X×X$ because then we have to get a binary algebra, i.e., the operation $X×X\to X$, which have to be commutative.
So the question (if these thoughts are right) is: how to map $X$ to $X×X$ to get later a commutative binary algebra?
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talhekh
Hint: Giving a function $f:A/\epsilon \phantom{\rule{thinmathspace}{0ex}}\to B$ from a quotient set $A/\epsilon$ is the same as giving a function $\overline{f}:A\to B$ that satisfies .
Apply this for $A=X×X$ and a suitable equivalence relation $\epsilon$ on it.