What is the name of this functor's property?

Question

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?

Answer 1

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.