Component functions of composition of mappings

Question

Given $\mathbf{F}:\mathcal{O}\to\mathbb{R}^m$ and $\mathbf{G}:\mathcal{U}\to\mathbb{R}^k$ where $\mathcal{O}$ and $\mathcal{U}$ are open subsets of $\mathbb{R}^n$ and $\mathbb{R}^m$ respectively, and also that $\mathbf{F}(\mathcal{O})$ is contained in $\mathcal{U}$, why is it the case that the composition $\mathbf{G}\circ\mathbf{F}$ can be written as $(G_1\circ\mathbf{F},G_2\circ\mathbf{F},\ldots,G_k\circ\mathbf{F})$, where $\mathbf{G}=(G_1,\ldots,G_k)$?

Answer 1

Since by definition $G_i \circ F = \pi_i \circ G \circ F = (G \circ F)_i$ for all $i \in \{1, \ldots, k\}$ what you wrote is equivalent to $((G \circ F)_1, (G \circ F)_2, \ldots, (G \circ F)_k)$