Coproducts in Eilenberg–Moore categories

Question

In Category Theory In Context, Proposition 5.6.11 Riehl says that for objects $(A_1,\alpha_1)$ and $(A_2,\alpha_2)$ in the Eilenberg–Moore category of monad $\mathcal{C} \xrightarrow{T} \mathcal{C}$ (for cocomplete $\mathcal{C}$) we have that the coproduct $(A,\alpha)$ of $(A_1,\alpha_1)$ and $(A_2,\alpha_2)$ equals the coequalizer of $$(T(TA_1 + TA_2),\mu_{TA_1 + TA_2}) \xrightarrow{T(\alpha_1 + \alpha_2)} (T(A_1 + A_2),\mu_{A_1+A_2})$$ and $$(T(TA_1 + TA_2),\mu_{TA_1 + TA_2}) \xrightarrow{T(\kappa)} (T^2(A_1+A_2),\mu_{T(A_1+A_2)}) \xrightarrow{\mu_{A_1+A_2}} (T(A_1 + A_2),\mu_{A_1+A_2})$$ where $T(A_1)+T(A_2) \xrightarrow{\kappa = [Ti_1,Ti_2]} T(A_1+A_2).$ My question is, how do I define the injection morphism from $(A_1,\alpha_1)$ into $(A,\alpha)$ that we should have if $(A,\alpha)$ is a coproduct?

An extra thing I’m wondering about (in case someone wants to help) is that if I have $(A_1,\alpha_1) \xrightarrow{f_1} (B,\beta)$ and $(A_2,\alpha_2) \xrightarrow{f_2} (B,\beta)$ what is the corresponding copairing $(A,\alpha) \xrightarrow{[f_1,f_2]} (B,\beta)$?

Answer 1

Each choice of a $T$-algebra structure $\beta\colon TB\to B$ establishes a bijection between pairs of morphisms $f_i\colon A_i\to B$ and $T$-algebra homomorphisms $(T(A_1+A_2),\mu_{A_1+A_2})\to (B,\beta)$ given by the unique factorization of the copairing morphism $\left<f_1,f_2\right>\colon A_1+A_2\to B$ through the unit $A_1+A_2\to T(A_1+A_2)$ followed by a $T$-algebra homomorphism $\left\{f_1,f_2\right\}\colon T(A_1+A_2)\to B$.

One can then prove Riehl's Proposition 5.6.11. (https://math.stackexchange.com/a/4549017/) by showing that this $T$-algebra homomorphism coequalizes the pair $T(\alpha_1+\alpha_2),\mu_{A_1+A_2}\circ T[Ti_1,Ti_2]\colon T(TA_1+TA_2)\to T(A_1+A_2)$ if and only if the $f_i\colon A_i\to B$ are $T$-algebra homomorphisms from $(A_i,\alpha_i)$ to $B$ .

This observation also answers both of your questions.

First, the coproduct $(A,\alpha)$ of the $T$-algebras has to be the coequalizer with coproduct inclusions of $T$-algebras the composites of the coproduct inlusions, unit, and coequalizer morphism: $A_i\to A_1+A_2\to T(A_1+A_2)\to A$.

Second, the copairing $T$-algebra homomorphism $[f_1,f_2]\colon A\to B$ is the unique factorization of the $T$-algebra homomorphism $\left\{f_1,f_2\right\}\colon T(A_1+A_2)\to B$ through the coequalizer $T(A_1+A_2)\to A$.