Functor generalisation In an article I am writing, I am led to the following generalization of the notion of functor. Let $C$ and $D$ and be two categories. A generalized functor $F : C \to D$ is given by:


*

*a function $f : C_0 \to D_0$

*for all $c \in C_0$, a set $F_c$

*for all $c_1,c_2 \in C_0$, a function $F_{c_1,c_2}:C(c_1,c_2) \times F_{c_2} \to F_{c_1} \times D(f(c_1),f(c_2))$.


All this must be compatible with the product and units, that is:


*

*for all $c \in C$ and all $x \in F_c$, $F_{c,c} (1_c, x) = (x,1_{f(c)})$

*for all $c_1,c_2,c_3 \in C$, the following composites are equal:


$$C(c_1,c_2) \times C(c_2,c_3) \times F_{c_3} \to C(c_1,c_3) \times F_{c_3} \to  F_{c_1} \times D(f(c_1),f(c_3)) $$
and
$$C(c_1,c_2) \times C(c_2,c_3) \times F_{c_3} \to C(c_1,c_2) \times  F_{c_2} \times D(f(c_2),f(c_3)) \to F_{c_1}\times D(f(c_1),f(c_2))\times D(f(c_2),f(c_3)) \to F_{c_1}\times D(f(c_1),f(c_3))$$
In particular, a functor is a generalized functor, where every $F_c$ is the singleton set.
Are you aware of a similar structure arising somewhere?
One way to see this structure is the following. One can see objects $a,b$ of a category as points, and then see $C(a,b)$ as an arrow form $a$ to $b$. If you say that the composition of two arrows is given by the Cartesian product, then you can see the composition in $C$ as a $2$-arrow from $C(a,b) \times C(b,c) \to C(a,c)$. 

Now how do you represent a functor in this setting? You can't represent the function $C(a,b) \to D(f(a),f(b))$ by a $2$-arrow, because the 'arrows' $C(a,b)$ and $D(f(a),f(b))$ are not parallel (the first one goes from $a$ to $b$, while the second goes from $f(a)$ to $f(b)$). 

One way to deal with that is to add two arrows: one called $F_{a}$ from $a$ to $f(a)$ and the other called $F_{b}$ from $b$ to $f(b)$. Now we have a square that we can fill with a $2$-arrow $F_{a,b}$. 

If we keep in mind the correspondence 'arrows are sets' and '$2$-arrows are functions' from the previous paragraph, we obtain the definition of a generalized functor.
Two more things to note:


*

*One can define generalized natural transformations between generalised functors. The usual naturality arrows $\eta_a : f(a) \to g(a)$ are then replaced by functions $G_a \to F_a \times D(f(a),g(a))$:





*

*This can be extended to $n$-functors by replacing the sets $F_a$ by $(n-1)$-categories and the functions $F_{a,b}$ by $(n-1)$-functors.

 A: Here is one way to look at it: if V is a monoidal category and $\mathbf{B} V$ is the corresponding one-object bicategory, then a V-category in the usual sense is the same thing as a lax functor $\mathrm{c}(C_0) \to \mathbf{B} V$, where $C_0$ is a set and $\mathrm{c}(\cdot)$ is the fully faithful functor that takes a set to the codiscrete category on it (i.e. the one with $C_0$ as object set and exactly one morphism between each pair of objects).  If C and D are V-categories then a V-functor from C to D is given by a function $f \colon C_0 \to D_0$ together with an identity-component oplax natural transformation (an icon)
 $\hat C \to \hat D \circ \mathrm{c}(f)$, where $\hat C, \hat D$ are the lax functors corresponding to C and D.  What you have seems to be a general oplax transformation, without the condition that its components be identities.  For that reason it's probably a not-unreasonable definition to write down, although I for one have never seen an example in the wild.
At the end of your question you are getting somewhere close to the notion of a category enriched in a bicategory, see here.
