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This seems fairly similar to density. Suppose I have three categories $A,B,C$, and a functor $L: B \to C$ so that every natural transformation $f: L.F \Rightarrow L.G$, for a parallel pair $F,G: A \to B$ belonging to some class of functors $\mathcal{F}$, factors as a natural transformation $g:F \Rightarrow G$ so that $L.g = f$. This feels like a thing that would come up in 2-category theory, but I'm struggling to find a good reference.

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    $\begingroup$ Could you clarify what your question is? It sounds like you're asking for $L$ to be fully faithful, so that postcomposition by $L$ induces an isomorphism between 2-cells $L \circ F \Rightarrow L \circ G$ and 2-cells $F \Rightarrow G$. $\endgroup$
    – varkor
    Commented Apr 20, 2021 at 23:13
  • $\begingroup$ @varkor full faithfulness would mean a unique factorisation $\endgroup$
    – David Roberts ♦
    Commented Apr 21, 2021 at 2:38
  • $\begingroup$ Yeah it’s a bit weaker than that - I have a 2-monad and three strict algebras of it. The morphism $B$ to $C$ acts like an inserter for every parallel pair of strict algebra homomorphisms. My example certainly isn’t a fully faithful functor - it’s actually a reflector. $\endgroup$
    – Ben MacAdam
    Commented Apr 21, 2021 at 3:39
  • $\begingroup$ I think it is a particular case of an inserter, but without uniqueness of the 1-cell; the non-uniqueness instead suggests some sort of (weak) orthogonality condition. $\endgroup$
    – fosco
    Commented Apr 21, 2021 at 5:58
  • $\begingroup$ If you don't want uniqueness of the factorisation, you can ask for $L$ just to be full. If it arises via a universal property (without uniqueness), then this is suggestive of a weak (2-)limit as @fosco points out. $\endgroup$
    – varkor
    Commented Apr 21, 2021 at 10:58

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