Fix a finitely well-complete category $\mathsf D$. Every strong factorization system $(\mathcal L,\mathcal R)$ such that $\mathcal L$ possesses the left cancellation property is known to induce a full replete reflective subcategory.
The action of the reflection $r$ on an object $D$ is the left factor of $D\to \bf 1$. I'm trying to work out the action of the reflection on a general arrow $f$ of $\mathsf D$, but I'm stuck. If $f\in \mathcal L$ then it's easy, but in the general case, there's a commutative diagram:
How to get the arrow $rX\to rD^\prime$?