Given a monoidal category $(\mathcal{C}, \otimes, I)$ with coproducts, the free monoid on an object $A \in \mathcal{C}$ is usually constructed by first constructing the free pointed object on $A$, i.e. $I + A$, and then constructing the free monoid over the pointed object $I + A$. The free monoid construction here is understood as constructing the free monoid from a pointed object.

Why is it not constructed following the inverse factorisation? Why not first construct the free semigroup and then adjoint a point to a semigroup? Is this approach followed in some paper?

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    $\begingroup$ How is the free monoid on $A$ the same as the free monoid on $I+A$ ? $\endgroup$ – Pece Jun 28 '18 at 9:57
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    $\begingroup$ @Pece It's the free monoid on $I+A$ as a pointed object, not just as an object $\endgroup$ – Kevin Carlson Jun 28 '18 at 23:47

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