Kelly's article uses factorization systems for constructing free monoids. What's the intuitive reading of factorization systems in this case? Also, for example, Barr in this article cites a result by Pavlovic:

Suppose that $\mathcal{V}$ has cofree coalgebras (commutative, associative, unitary) for its tensor product and has complete subobject lattices. Then for a factorization system $\mathcal{E}/\mathcal{M}$ satisfying the condition that $e \in \mathcal{E}$ implies $e \multimap K \in \mathcal{M}$, the category $Chu_{se}(\mathcal{V}, K)$ has cofree coalgebras as well.

I can read the formal definition of factorization system, but I do not see why they end up being so important for the construction of the free structures, or the role in that theorem by Pavlovic. The factorization systems seem to come from nowhere, and have no relation with the monoidal structures. Is there any intuition for understanding this?


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