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For $(A,a)$, consider the reflexive free-algebra presentation of a monad algebraBoth parallel morphisms are monad algebra morphisms: ${\mu }_A$ satisfies the monad associativity law, while $Ta$ is the image under $F^T$ of $a$. The map $a$ is a monad algebra morphism because $a{\mu }_A=aTa$, and the same equation says it equalizes the pair.

Suppose $h:F^T(A)\to (B,b)$ is a monad algebra morphism with $h{\mu }_A=hTa$. Put $k=h{\eta }_A:A\to B$. Naturality of $\eta$ and the unit law giveMoreover, since $h$ is a monad algebra morphism,Thus $k$ is a monad algebra morphism and factors $h$ through $a$. If $k^{\prime }a=h$, then $k^{\prime }=k^{\prime }a{\eta }_A=h{\eta }_A=k$, proving uniqueness. Therefore every monad algebra is the stated coequalizer of free algebras.

The underlying fork is a split coequalizer: take $q=a$, $s={\eta }_A$ and $t={\eta }_{TA}$. Then $qs=1_A$, ${\mu }_{A}t=1_{TA}$, and $(Ta)t=sa$. Its splitting maps need not be monad algebra morphisms; the preceding argument separately proves the coequalizer in ${\mathcal{C}}^T$.

For an algebra for a monad $(A,a)$, consider the fork in the Eilenberg-Moore categoryHere ${\mu }_A:F(TA)\to F(A)$ is the counit at $F(A)$, and $Ta=F(a)$. The arrow $a$ is an algebra morphism by $a{\mu }_A=aTa$, which also says that it coequalizes the two arrows.

Let $h:F(A)\to (B,b)$ be an algebra morphism with $h{\mu }_A=hTa$. Define $g=h{\eta }_A:A\to B$. Naturality of $\eta$ at $a$ givesSince $h$ is an algebra morphism,Thus $g:(A,a)\to (B,b)$ is an algebra morphism with $ga=h$. Any other such factorization satisfies $g^{\prime }=g^{\prime }a{\eta }_A=h{\eta }_A=g$. This proves the full coequalizer universal property inside the algebra category.

The pair is moreover a reflexive pair: its common section is $F({\eta }_A)$, with underlying map $T{\eta }_A$, because ${\mu }_{A}T{\eta }_A=1_{TA}$ and $TaT{\eta }_A=T(a{\eta }_A)=1_{TA}$. Hence every monad algebra is a reflexive coequalizer of free algebras. This reflexive free-algebra presentation of a monad algebra needs no general existence theorem for arbitrary algebra-category colimits.