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in [G] p.29, J.W Gray define the 2-comma category $[F, G]$ of two 2-functors $F: \mathcal{A}\to \mathcal{D},\ G: \mathcal{B}\to \mathcal{D}$. This definition work well also if we suppose $F$ a lax-functor and $G$ a op.lax.functor (in the former definition, about the composition rectify composing by cells $F(g) F(f)\to F(gf)$ and $G(gf)\to G(gf)$ ). If we suppose that $F$ and $G$ are normal, then the natural 2-functor projection $P: [F, G]\to \mathcal{A}\times \mathcal{B}$ make $[F, G]$ a span-(2-)fibration from $\mathcal{A}$ to $\mathcal{C}$ (see [J] p. 514 or [MB]), the canonical direct image (opcartesian lifting) or inverse image (cartesian lifting) are given by mere composition and compiling a commutative square by identities.

Consider a couple of 2-functors $F: \mathcal{A}\to \mathcal{C},\ G: \mathcal{C}\to \mathcal{A}$. In [G] p.174 J.W: Gray claim (as corollary from the theory developed in its Cap. 5) the there is a natural bijection between the 2-functors $S: [F, \mathcal{C}]\to [\mathcal{A}, G]$ that preserving the canonical op.clivage (direct image) and the lax.transformation (Gray call these quasi-transformation) $\eta: 1\Rightarrow G\circ F$ (this is also a pretty exercise).

Now what happen if we try to generalize this for $F$ lax.functor (normal), $G$ oplax.functor (normal)?

The first thing that come to eyes is $G\circ F$ isn't defined! (lack of consistency about the composition of morphisms).

Anyway if we do the some steps of the exercise above we find:

1) $\eta:A: A\to GF(A)$, such that $S(h)=G(h)\circ \eta_A$ for $h: F(A)\to X$: consider $(1_A,1, h): (A, 1_{F(A)}, F(A))\to (A, h, X)$ and that $S$ preserve the op.cartesian lifting as it is the above morphism.

2) $\eta_f: GF(f)\circ \eta_A \Rightarrow \eta_B \circ f$ for $f\in \mathcal{A}(A, B)$: consider $(f, 1, F(f)): (A, 1_{F(A)}, F(A))\to (B, 1_{F(B)}, F(B))$ .

3) and for $g\circ f: A\to B\to C$ in $\mathcal{A}$ we have that the composition of obvious cell $G(Fg\circ Ff)\to GF(g\circ f)\to g\circ f$ is equal to $\eta_{g\circ f}$: consider the composition of $(f, 1, F(f)): (A, 1_{F(A)}, F(A))\to (B, 1_{F(BA)}, F(B))$ and $(g, 1, F(g)): (B, 1_{F(B)}, F(B))\to (C, 1_{F(C)}, F(C))$.

We have the obvious cell $G(F_{g, f}):G(Fg\circ Ff)\to GF(g\circ f)$ and $G_{Fg, Ff}:G(Fg\circ Ff)\to GF(g)\circ GF(f)$.

this suggest to define the notion of bilax.funtor (normal) $H: \mathcal{D}\to \mathcal{E}$ as a morphism of $CAt$-graphs, such that for $g\circ f: A\to B\to C$ in $\mathcal{D}$ we have a morphism $[H; g, f]: H(A)\to H(C)$ natural in $g$ and $f$ and canonical cell's $H^{g, f}: [H; g, f]\to Hg\circ Hf$ and $H_{g, f}: [H; g, f]\to H(g, f)$, and respect to $H^{g, f}$ we require that $H$ is (in the obvious generalization form) a op.lax.funtor, analogously respect $H_{g, f}$ we require that $H$ is a lax.functor, and some other coherence law's between these two structure.

EDIT (exat definition of bilax.functor):

Define a \textbf{bilax.funtore} (normal) $H: \mathscr{A}\to \mathscr{C}$ as a $CAt$-graph morphism with $H(1_A)=1_{H(A)}$ and such that:

1) Given $h\circ g\circ f: A\to B\to C\to D$ in $\mathscr{A}$ is assigned $[h, g, f]: H(A)\to H(D)$ and given $g\circ f: A\to B\to C$ is assigned $[g, f]: H(A)\to H(C)$. These two maare natural respect the morphisms $f$, $g$, and (eventually) $h$.

2) Are assigned natual cell's: $[h, g, f]\to H(h)\circ [g, f]$, $[h, g, f]\to [h, g]\circ H(f)$ and $[h, g, f]\to [h\circ g, f]$, $[h, g, f]\to [h, g\circ f]$ and $[g, f]\to H(g)\circ H(f)$, $[g, f]\to H(g\circ f)$.

3) We have the commutative diagram (I write the latex code delete the global \textb{}):

\text{ \xymatrix{ [h,g, f]\ar[d]\ar[r]&[h\circ g, f]\ar[d]\ [h, g\circ f]\ar[r]& H(h\circ g\circ f) }\ ,\ \xymatrix{ [h,g, f]\ar[d]\ar[r]&[h,g]\circ H(f)\ar[d]\ H(h)\circ [g, f]\ar[r]& H(h)\circ H(g)\circ H(f)) } }

We have that $H$ is a lax.funtore exactly if $[h, g, f]= H(h)\circ H(g)\circ H(f)$, $[g, f]= H(g)\circ H(f)$ and the above morphism $[g, f]\to H(g)\circ H(f)$ is the identity, and it is a oplax.functor exatly if $[h, g, f]= H(h\circ g\circ f)$, $[g, f]= H(g\circ f)$ and $[g, f]\to H(g\circ f)$ is the identity.

Come back above, we have that $G\circ F$ is a bilax.funtor and $\eta: 1\to G\circ F$ a lax.tranformatin between oplax.functors (considering the "oplax-face" of $G\circ F$).

I ask if this is just know in literature.

[G]: John W. Gray, Formal category theory: adjointness for 2 2 -categories. Lecture Notes in Mathematics, Vol. 391

[J]: B. Jacobs, Categorical Logic and Type Theory, Studies in Logic and the Foundations of Mathematics 141, North Holland, Elsevier, 1999.

[MB] Marta Bunge, Bifibration-Induced Adjoint Pairs. in: J. Gray (editor), Reports of the Midwest Category Seminar V - Zurich 1970. Lecture Notes in Mathematics 195 (1971) 70-121.

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