Let $(F, \phi): \mathscr{A}\to \mathscr{B}$ a lax-functor between 2-categories. In the setting of 2-categories the axiom of lax-functor become:

**Ul)** $1: F(f)=F(f)\circ 1_{F(X)}\xrightarrow{1\circ \phi_X}F(f)\circ F(1_X)\xrightarrow{\phi_{f, 1}} F(f\circ 1_X)= F(f)$.

For $f: X \to Y$ in $\mathscr{A}$.

**Ur)** $1: F(f)= 1_{F(Y)}\circ F(f)\xrightarrow{\phi_X\circ 1} F(1_Y)\circ F(f)\xrightarrow{\phi_{1, f}} F( 1_Y \circ f)= F(f)$.

For $f: X \to Y$ in $\mathscr{A}$.

**UA)** The compositions $F(h)\circ F(g)\circ F(f) \xrightarrow{\phi_{h, g}\circ 1} F(h\circ g)\circ F(f)
\xrightarrow{\phi_{h\circ g, f}\circ 1} F(h\circ g\circ f)$ and

$F(h)\circ F(g)\circ F(f) \xrightarrow{1\circ \phi_{g, f}} F(h)\circ F(g\circ f) \xrightarrow{\phi_{h, g\circ f}} F(h\circ g\circ f)$ are equal.

For componibile morphisms $h, g, f$.

.

I call $(F, \phi)$ normal (or unitary) if in the axioms $(Ul)$ and $(Ur)$ above all arrows are identities.

Gived $(F, \phi)$ (general) I define a normal lax.functor $(\tilde{F}, \tilde{\phi})$ that is the some of $F$ on objects and on non-identity morphisms, with of course $\tilde{F}(1_X)=1_{F(X)}$, $\widetilde{\phi}_{X}= 1: \tilde{F}(1_X)\to 1_{F(X)} $, and with
$\tilde{\phi}_{g, f}$ defined as:

$ \phi_{g, f}: F(g)\circ F(f)\to F(g\circ f)$ if $f$ and $g$ arent identity, and the obvious identity if $f$ or $g$ is a identity. I checked (easly) that axiom $(UA)$ is true for $(\tilde{F}, \tilde{\phi})$, then $(\tilde{F}, \tilde{\phi})$ is a lax.funtor.

I ask if this (very easy normalization) is just know in literature ( I dont know), and if is right (I'm sure its right, but I'm no too sure of myself).

**Edit:** If in $(Ul)$ all arrows are identies, this imply the some in $(Ur)$ (i.e. $(F, \phi)$ is normal)?

My motivations is a generalization of the J.W. Gray concept of quasi-functors to lax.functors (Gray gived this definition for 2-functors).

**EDit** As Jonathan Chiche observed, these is the obvious condition than a lax.functor induce hom-functors between hom-categories (I am ashamed for this rough oversight). Anyway if the canonical morphism $\phi_X: 1_{FX}\Rightarrow F(1_X)$ is a isomorphism then the mine definition preserving this funtorialiy, only need a easy correction based on the following observation: the cells of type $\sigma: 1_{FX}\Rightarrow W$ are in bijections with the cells of type $\sigma': F(1_X)\Rightarrow W$ and the cells like $\tau: W\Rightarrow 1_{FX}$ are in bijections with the cells of type $\tau': W\Rightarrow F(1_X)$. With this correction I seems that also the naturality of $\tilde{\phi}$ work well.