Reflexive free-algebra presentation of a monad algebra ($F(TA)\rightrightarrows F(A)\to (A,a)$)

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.