Let $f, g : [0,1]^2 \to \mathbb{R}$ be two smooth functions, which are strictly increasing and concave in each coordinate. That is, for every $0 < x,y < 1$ we have $\frac{\partial f}{\partial x}(x,y) > 0$, $\frac{\partial f}{\partial y}(x,y) > 0$, $\frac{\partial^2 f}{\partial x^2}(x,y) < 0$, $\frac{\partial^2 f}{\partial y^2}(x,y) < 0$, and the same holds for $g$.

I am interested in solutions (in $[0,1]^2$) of the system of equations:

$$\frac{\partial f}{\partial x}(x,y) = \frac{\partial g}{\partial x}(1-x,1-y),$$ $$\frac{\partial f}{\partial y}(x,y) = \frac{\partial g}{\partial y}(1-x,1-y).$$

I would like to know under which conditions on $f$ and $g$ this system has a unique solution.

[Edited on 29-March-2020 to make the question clearer]

Let $f, g : [0,1]^2 \to \mathbb{R}$ be two smooth functions, which are strictly increasing and concave in each coordinate. That is, for every $0 < x,y < 1$ we have $\frac{\partial f}{\partial x}(x,y) > 0$, $\frac{\partial f}{\partial y}(x,y) > 0$, $\frac{\partial^2 f}{\partial x^2}(x,y) < 0$, $\frac{\partial^2 f}{\partial y^2}(x,y) < 0$, and the same holds for $g$.

A point $(x_0,y_0) \in [0,1]^2$ is a solution if $$\frac{\partial f}{\partial x}(x_0,y_0) = \frac{\partial g}{\partial x}(1-x_0,1-y_0),$$ $$\frac{\partial f}{\partial y}(x_0,y_0) = \frac{\partial g}{\partial y}(1-x_0,1-y_0).$$

I would like to know under which conditions on $f$ and $g$ this system has a unique solution.