Final Topology Contains Codomain Topology iff Mappings are Continuous

Theorem
Let $\struct{Y, \tau}$ be a topological space.

Let $\family {\struct{X_i, \tau_i}}_{i \mathop \in I}$ be a family of topological spaces.

Let $\family {f_i}_{i \mathop \in I}$ be a family of mappings $f_i : X_i \to Y$.

Let $\tau'$ be the final topology on $Y$ with respect to $\family {f_i}_{i \mathop \in I}$.

Then:
 * $\tau \subseteq \tau'$ $\forall i \in I : f_i: \struct{X_i, \tau_i} \to \struct{Y, \tau}$ is continuous.