= Reflexive free-algebra presentation of a monad algebra
{title2=$F(TA)\rightrightarrows F(A)\to(A,a)$}
Every <algebra for a monad> $(A,a)$ is the <coequalizer>, in the <Eilenberg-Moore category>, of $F(TA)\rightrightarrows F(A)$ with underlying arrows $\mu_A$ and $Ta$, followed by $a:F(A)\to(A,a)$. The pair has common section $F\eta_A$. If an algebra morphism $h:F(A)\to(B,b)$ equalizes the pair, the unique induced algebra morphism is $h\eta_A:(A,a)\to(B,b)$. This explicit proof does not require general colimits of algebras.
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