Suppose that $X$, $Y$ and $Z$ are finite vector spaces of dimension $k$, $l$ and $m$ such that $x\in X$ , $y\in Y$ and $g(x,y)\in Z$. I define a function $\pi:X\times Y\to X\times Y \times Z$ such that
$$\pi(x,y)=\sigma\circ\left( 1_X\times 1_Y \times g(x,y)\right)$$
where $1_X$ is the identity on $X$ and $1_Y$ the identity on $Y$ and $1_X\times 1_Y \times g(x,y):X\times Y\to X\times Y\times Z$ namely $(x,y)\to(x,y,g(x,y))$ where every $(x,y)$ is associated with one $g(x,y)$, namely $g$ is a bijective (and hence injective) function. For simplicity we further assume that the dimension of the vector space $X\times Y$ is $N$. By writing $\pi_i(x,y)$ we refer to the $i$-th vector of $\pi$ such that $\pi_i=\sigma_i\circ\left( 1_X\times 1_Y \times g_i(x,y)\right)$
I want to define a mapping from the space $X\times Y \times Z_i$ to the $j$-th coordinate $z_j\in Z_i$. I define $(\tau_j)_{j\in I}:X\times Y \times Z_i\to Z_j$ such that
$$\tau_j(x,y,z_i)=pr_j\circ\sigma^{-1}_i$$
If $\sigma_i$ serves as a kind of a function that could change the order of the vector $\left(x,y,g_i(x,y)\right)$, could this be a permutation, namely
$$\sigma_i=\begin{pmatrix}(x_1,y_1) & (x_2,y_2) & \cdots & (x_j,y_j) & \cdots & (x_N,y_N)\\ g_1(x_1,y_1) & g_2(x_2,y_2) & \cdots & g_j(x_j,y_j) & \cdots & g_N(x_N,y_N)\end{pmatrix}$$
where $\tau_j$ could return to us only the $j$-coordinate? Is it well defined?