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Assume there is a functor $L$ from a category $C$ to a category $D$ which satisfies the following property: for any objects $X,Y,Z$ from $C$ and morphisms $f\colon X\to Y, g\colon X\to Z$ such that $L(g)=\varphi\circ L(f)$ for some morphism $\varphi\colon L(Y)\to L(Z)$ there is a morphism $h\colon Y\to Z$ such that $\varphi=L(h)$ and $g=h\circ f$. What is the name of this property?

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  • $\begingroup$ Do you require uniqueness of $h$? $\endgroup$ Commented Mar 19 at 8:38
  • $\begingroup$ Both cases are ok for me $\endgroup$
    – 8k14
    Commented Mar 19 at 8:48
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    $\begingroup$ In model theory the dual of this property ($L(g) = L(f) \circ \varphi$ implies there is $h$ s.t. $L(h) = \varphi$ and $g = f \circ h$) is sometimes called "coherence". So ...... "herence"? $\endgroup$ Commented Mar 19 at 9:11
  • $\begingroup$ @Mark Kamsma Thanks. Thus "coherence" of $L$ is equivalent to the fact that all the morphisms in $C$ are $L$-cartesian? $\endgroup$
    – 8k14
    Commented Mar 19 at 9:51
  • $\begingroup$ @8k14 It seems so, though I have not seen the uniqueness requirement before (and I must admit that I had never heard of the term "$L$-cartesian"), but it would be automatic in most model-theoretic contexts. Usually the model-theoretic context is where $L$ is an inclusion of categories (not necessarily full, of course). Even more specifically: $D$ could be thought of as the category of structures with embeddings and $C$ the category of models of some theory with elementary embeddings, and $L$ is then the obvious inclusion. $\endgroup$ Commented Mar 19 at 11:24

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In case we require this $h$ to be unique, then the property that you wrote down is saying that $f$ is an $L$-cocartesian morphism (see Definition 2.1 here), so $L$ is a functor for which every morphism in $C$ is $L$-cocartesian. On its own, this does not bring us much (except that all fibers of $L$ are groupoids), but we could additionally ask for the following: for every morphism $a:A\to B$ in $D$ and every object $A'$ in $C$ such that $LA'=A$, there exists a morphism $a'\colon A'\to B'$ in $C$ such that $La'=a$. If this is also satisfied, then your functor $L$ is a Grothendieck opfibration fibered in groupoids, also called a left fibration in more modern higher category theory. (By the Grothendieck construction, this would mean that $L$ classifies a $(2,1)$-functor $D\to\mathsf{Grpd}$, where $\mathsf{Grpd}$ is the $(2,1)$-category of groupoids. Informally, this functor is of the form $d\mapsto L^{-1}(d)$.)

If we do not require $h$ to be unique, then I am not aware of any definition that talks about this.

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  • $\begingroup$ Thanks so much. My functor does not satisfy the extra property you describe but every morphism in $C$ is also $L$-cartesian. Does it give anything? $\endgroup$
    – 8k14
    Commented Mar 19 at 9:45
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    $\begingroup$ Not on its own, as far as I know. Which functor are you exactly considering? It does hold that a Grothendieck opfibration that is also a Grothendieck fibration with the one-arrow category $0\to 1$ as target encodes an adjunction, so maybe you can restrict $D$ to a subcategory on which $L$ is such a functor, and then we would get a whole bunch of adjunctions encoded by $L$. $\endgroup$ Commented Mar 19 at 17:10

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