# The reflection of a reflective factorization system

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 $rX\to rD^\prime$?

• Use the orthogonality property of a factorization system for $\lambda_D$ and $\rho_{D'}$. – Mike Shulman Aug 3 '16 at 3:21
• @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. – Arrow Aug 3 '16 at 18:58

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