Let $P: \mathcal{C}\to \mathcal{A}$ be a functor. We call a morphism $v$ of $\mathcal{C}$ *vertical* (over $A\in \mathcal{A}$) if $P(v)=1_A$; then we have the fibre category $\mathcal{C}_A$ of vertical morphisms over $A$.
We call a morphism $f: X\to Y$ in $\mathcal{C}$ *cartesian* if for $f': X'\to Y$ with $P(f)=P(f')$, there exists a unique vertical morphism $v: X'\to X$ such that $f'=f\circ v$.

If $f: A\to B$ is in $\mathcal{A}$ and $Y\in \mathcal{C}(B)$, we call a cartesian morphism of type $\theta_f(Y): f^\ast Y\to Y$ with $P(\theta_f)=f)$ a *cartesian $f$-lifting* of $Y$. If there exists such for each $f$ and $Y$, we call $P$ a *prefibration*, and a *precleavage* is a choice of $\theta_f(Y)$ for each $f$ and $Y$. If we choose $\theta_{1_A}=1_A$, the preclivage is called *normal*.

After fixing a cleavage, we obtain a functor $\theta_f: \mathcal{C}(B)\to \mathcal{C}(A)$, and for $X\xrightarrow{f}Y\xrightarrow{g}Z$ there is a transformation $c_{g, f}: f^\ast\circ g^\ast\Rightarrow (g\circ f)^\ast$ (considering $ \theta_g(Z)\circ \theta_f(g^\ast(Z))$). Thus, we have a lax functor $\textbf{C}: \mathcal{A}^{op}\to CAT$ defined as $\textbf{C}(A)=\mathcal{C}(A)\ A\in\mathcal{A}$, $\textbf{C}(f)=f^{\ast}$,

$C_{g,f}=c_{g, f}: \textbf{C}(f)\circ \textbf{C}(g)\Rightarrow \textbf{C}(g\circ f)$.

Let $P: \mathcal{C}\to \mathcal{A}$ and $Q: \mathcal{D}\to \mathcal{A}$ be prefibrations with fixed normal cleavages. If $T: \mathcal{C}\to \mathcal{D}$ is a functor over $\mathcal{A}$, i.e. with $Q\circ T= P$, then by restriction we have functors $T_A: \textbf{C}(A)\to \textbf{D}(A)$. Applying $T$ to $\theta_f(Y)$ and considering the cartesian $\theta_f(T(Y))$ (in $\mathcal{D}$), we have a natural vertical morphism $T_f(Y): T(f^\ast(Y))\to f^\ast(T(Y))$, which defines a transformation $T_f: T_A\circ \textbf{C}(f)\Rightarrow \textbf{D}(f)\circ T_B$. The data $(T_A, T_F)_{A, f}$ is what is called an oplax transformation. In the other direction, from an oplax transformation $(T_A, T_F)_{A, f}$ we can make a functor $T: \mathcal{C}\to \mathcal{D}$ over $\mathcal{A}$ (considering that each morphism over $\mathcal{C}$ or in $\mathcal{D}$ has a unique factorization as a vertical morphism followed by a cartesian morphism).

**Thus the oplax transformations are identified with functors $T$ over $\mathcal{A}$**.

**My question is**:

If (instead of an **oplax tranformation**) we consider a **lax tranformation** $T$, i.e. a family

$T_A: \textbf{C}(A)\to \textbf{D}(A)$ and $T_f: \textbf{D}(f)\circ T_B \Rightarrow T_A\circ \textbf{C}(f)$

with the coherence conditions:

$T_{gf}\ast (d_{g, f})\circ T_C)=$

$(T_A\circ c_{g, f})\ast(T_f\circ \textbf{C}(g))\ast(\textbf{D}(f)\circ T_g): \textbf{D}(f)\circ\textbf{D}(g)\circ T_C\Rightarrow T_A\circ \textbf{D}(f)\circ\textbf{D}(g)$

(the unitary condition being unnecessary, by our choice of normal cleavages).

**Is there some way to represent $T$ as a categorical construction?**