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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:

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How to get the arrow $rX\to rD^\prime$?

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    $\begingroup$ Use the orthogonality property of a factorization system for $\lambda_D$ and $\rho_{D'}$. $\endgroup$
    – Mike Shulman
    Commented Aug 3, 2016 at 3:21
  • $\begingroup$ @MikeShulman thanks and sorry for the trivial question. If you could post this as an answer (even if only for the sake of not leaving it unanswered) I will gladly accept. $\endgroup$
    – Arrow
    Commented Aug 3, 2016 at 18:58

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Use the orthogonality property of a factorization system for $\lambda_D$ and $\rho_{D'}$.

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