This is a reformulation of a Mathematics Stack Exchange question that got no answer there, and, at the same time, a follow-up to another question on MSE.

The original problem was to prove $U(\mathfrak{g}\oplus \mathfrak{h})\cong U(\mathfrak{g})\otimes U(\mathfrak{h})$, where $U(\mathfrak{g})$ is the universal enveloping algebra of a Lie algebra $\mathfrak{g}$ using universal properties. To do that, the person answering the question made the following two statements:

- $\otimes$ is the commutative coproduct in the category of (associative, unital) $\mathbb{K}$-algebras, where $\mathbb{K}$ is a field.
- $\oplus$ is the commutative coproduct in the category of Lie algebras.

Both objects were defined as such that satisfy the diagram for coproduct only for a specific subset of morphisms from cofactors to the third object, namely:

- $R \otimes S$ satisfies the coproduct diagram for every $A$ and every pair $\sigma : R \to A, \tau : S \to A$ such that $\forall x \in R, y \in S: \sigma(x)\tau(y)=\tau(y)\sigma(x)$ in $A$.
- $\mathfrak{g} \oplus \mathfrak{h}$ satisfies the coproduct diagram for every $\mathfrak{a}$ and pair $\tau: \mathfrak{g} \to \mathfrak{a}, \sigma : \mathfrak{h} \to \mathfrak{a}$ such that $\forall x \in \mathfrak{g}, y \in \mathfrak{h}: [\sigma(x), \tau(y)]=0$ in $\mathfrak{a}$.

Using these statements, the person was able to show that $U(\mathfrak{g}\oplus \mathfrak{h})\cong U(\mathfrak{g})\otimes U(\mathfrak{h})$. What got my attention is that their first attempt at the proof was based on the assumption that $\otimes$ is just the coproduct in the category of associative unital algebras, but after an objection from my side that $\otimes$ is only a coproduct in the category of commutative associative unital rings, the person modified their original proof by introducing these new objects.

Naturally, I am curious if this is a coincidence, that an incorrect proof assuming an object is a coproduct for a non-abelian category, while in reality it is only a coproduct for the commutative part of the category, could be turned into a correct proof that uses commutative coproduct in non-abelian category. Is here some adjunction at play?

But what intrigues me the most is that I haven't been able to find a single mention of the commutative coproduct on the Web. So, what is this thing, generally, and where can I learn more about it?