Characteristic property of the disjoint union topology

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Finish proof at least, link to other pages

Statement

Let $\big((X_\alpha,\mathcal{J}_\alpha)\big)_{\alpha\in I}$ be a collection of topological spaces and let $(Y,\mathcal{ K })$ be another topological space]]. We denote by $\coprod_{\alpha\in I}X_\alpha$ the disjoint unions of the underlying sets of the members of the family, and by $\mathcal{J}$ the disjoint union on it (so $(\coprod_{\alpha\in I}X_\alpha,\mathcal{J})$ is the disjoint union topological construct of the $\big((X_\alpha,\mathcal{J}_\alpha)\big)_{\alpha\in I}$ family) and lastly, let $f:\coprod_{\alpha\in I}X_\alpha\rightarrow Y$ be a map (not necessarily continuous) then:
we state the characteristic property of disjoint union topology as follows:

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TODO: rewrite and rephrase this

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• $f:\coprod_{\alpha\in I}X_\alpha\rightarrow Y$ is continuous if and only if $\forall\alpha\in I\big[f\big\vert_{X_\alpha^*}:i_\alpha({X_\alpha})\rightarrow Y\text{ is continuous}\big]$

Where (for $\beta\in I$) we have $i_\beta:X_\beta\rightarrow\coprod_{\alpha\in I}X_\alpha$ given by $i_\beta:x\mapsto(\beta,x)$ are the canonical injections

Proof

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It's actually pretty easy, just got to be careful with the notation