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once I typed out at category of monoids some details of the tedious construction of pushouts in $Monoids(C)$ (for $C$ a symmetric monoidal category) along a morphism of free monoids
$\array{ F(K) &\to& A \\ {}^{\mathllap{F(v)}}\downarrow && \downarrow \\ F(L) &\to& A \coprod_{F(K)} F(L) }$for some morphism $v : K \to L$ in the underlying category $C$.
I remember when typing this I thought I knew how this simplifies in the case of commutative monoids. But now I come back to this, find that I forgot what I knew and need to think again.
Is in $CommMonoids(C)$ the pushout of the above kind given by the pushout in the underlying category $C$?
oh, I am being stupid. For instance page 478 of the Elephant has what I need. I’ll write out something into the $n$Lab, lest I forget again.
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