Component Maps of a Map into $\mathbb{R}^{q}$, Entry Maps of a Map into Matrices, and Maps Assembled from Them

In the setting of Euclidean Space and Real Matrices: Standing Notation, let $p,q\in\mathbb{N}$ and let $S$ be a set, with $\mathbb{R}^{S}$ the set of maps from $S$ to $\mathbb{R}$. For $z\in\mathbb{R}^{q}$ and $k\in[q]$, $z_{k}$ is the $k$-th component of $z$, and for $B\in\mathcal{M}_{p\times q}(\mathbb{R})$, $i\in[p]$ and $j\in[q]$, $B_{ij}$ is the entry of $B$ in row $i$ and column $j$, as in Euclidean Space and Real Matrices: Standing Notation (ii) and Euclidean Space and Real Matrices: Standing Notation (iii). A map $y:[q]\to\mathbb{R}^{S}$ is written $k\mapsto y_{k}$, with $y_{k}=y(k)$, and a map $b:[p]\times[q]\to\mathbb{R}^{S}$ is written $(i,j)\mapsto b_{ij}$, with $b_{ij}=b(i,j)$.

(i) Let $x:S\to\mathbb{R}^{q}$ be a map. For every $k\in[q]$ there is exactly one map $x_{k}:S\to\mathbb{R}$ with $x_{k}(u)=x(u)_{k}$ for every $u\in S$; and there is exactly one map from $[q]$ to $\mathbb{R}^{S}$ whose value at every $k\in[q]$ is $x_{k}$, written $k\mapsto x_{k}$.

(ii) Let $A:S\to\mathcal{M}_{p\times q}(\mathbb{R})$ be a map. For all $i\in[p]$ and $j\in[q]$ there is exactly one map $A_{ij}:S\to\mathbb{R}$ with $A_{ij}(u)=A(u)_{ij}$ for every $u\in S$; and there is exactly one map from $[p]\times[q]$ to $\mathbb{R}^{S}$ whose value at every $(i,j)\in[p]\times[q]$ is $A_{ij}$, written $(i,j)\mapsto A_{ij}$.

(iii) For every map $k\mapsto y_{k}$ from $[q]$ to $\mathbb{R}^{S}$ there is exactly one map $x:S\to\mathbb{R}^{q}$ with $x(u)_{k}=y_{k}(u)$ for all $u\in S$ and $k\in[q]$, and for this map, with $x_{k}$ as in (i), $x_{k}=y_{k}$ for every $k\in[q]$.

(iv) For every map $(i,j)\mapsto b_{ij}$ from $[p]\times[q]$ to $\mathbb{R}^{S}$ there is exactly one map $A:S\to\mathcal{M}_{p\times q}(\mathbb{R})$ with $A(u)_{ij}=b_{ij}(u)$ for all $u\in S$, $i\in[p]$ and $j\in[q]$, and for this map, with $A_{ij}$ as in (ii), $A_{ij}=b_{ij}$ for all $i\in[p]$ and $j\in[q]$.