Is there any literature on representable presheaves which are fibrations, or categories such that all representable presheaves are fibrations?

A representable presheaf $$\mathcal{C}(-,X):\mathcal{C}^{op}\to{\bf Set}$$ being a fibration would mean that for any function $$f':Y'\to \mathcal{C}(A,X)$$ there exists a Cartesian arrow $$f:A\to Y$$ in $\mathcal{C}$ such that $$\mathcal{C}(-,X)(f)=\circ f=f'$$ so $f'$ is just precomposition with $f$, and in particular $Y'=\mathcal{C}(Y,X)$. This in turn means that for any coinitial arrow $$g:A\to Z$$ (coterminal in $\mathcal{C}^{op}$) such that there exists a function $v':\mathcal{C}(Z,X)\to\mathcal{C}(Y,X)$ satisfying $$\mathcal{C}(-,X)(f)\circ v'=\mathcal{C}(-,X)(g)$$ there exists a unique arrow $$v:Y\to Z$$ such that $v'$ is precomposition with $v$ and $$v\circ f=g.$$

All of this seems to be saying that we have nice representability properties for functions into Hom-sets such that the representable presheaf of the codomain object in the Hom-set is a fibration. This seems like something interesting, but a brief search for literature only turned up the notion of a fibrant object in a model category.

The Yoneda embedding of a category $\mathcal{C}$ being a fibration also yields a nice representability condition for natural transformations $\alpha:F\Rightarrow\mathcal{C}(-,X)$ from arbitrary presheaves into representable ones ($F$ is also strictly representable, and $\alpha$ is just pointwise postcomposition with some arrow in $\mathcal{C}$), but the universal property of Cartesian arrows is immediately satisfied for all arrows in $\mathcal{C}$ under the Yoneda embedding by faithful fullness of Yoneda, so the universal property isn't adding anything new unless I'm mistaken. Any references or insights are appreciated.