Partial Iterates of a Two-Variable Map, Row by Row and Column by Column

In the setting of Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices, let $\ast$ be a binary operation on a set $X$, let $A$ and $B$ be sets, and for $N\in\mathbb{N}$ let $[N]$ be as in Intervals of Natural Numbers (ii). For a map $f$ on a Cartesian product, $f(x,y)$ stands for $f((x,y))$, as in Sets and Maps: Ordinary Notation (v). For $N\in\mathbb{N}$ and an expression $t(n)$ in $X$ for $n\in[N]$, $\mathop{\ast}\limits_{n=1}^{N}t(n)$ is the iterated operation of $n\mapsto t(n)$ on $[N]$.

(i) For every map $f:A\times\mathbb{N}\to X$ there is exactly one map $g:\mathbb{N}\times A\to X$ with $g(N,x)=\mathop{\ast}\limits_{n=1}^{N}f(x,n)$ for all $N\in\mathbb{N}$ and $x\in A$.

(ii) For every map $f:\mathbb{N}\times A\to X$ there is exactly one map $g:\mathbb{N}\times A\to X$ with $g(N,y)=\mathop{\ast}\limits_{m=1}^{N}f(m,y)$ for all $N\in\mathbb{N}$ and $y\in A$.

(iii) For every map $f:A\times B\to X$ there is exactly one map $h:B\times A\to X$ with $h(y,x)=f(x,y)$ for all $x\in A$ and $y\in B$.