Homeomorphic Topology of Initial Topology is Initial Topology

Theorem
Let $\struct{X_\alpha, \tau_\alpha}, \struct{X_\beta, \tau_\beta}$ be topological spaces.

Let $\family {\struct {Y_i, \tau_i} }_{i \mathop \in I}$ be an indexed family of topological spaces where $I$ is an arbitrary index set.

Let $\family {f_i: X_\beta \to Y_i}_{i \mathop \in I}$ be an indexed family of mappings indexed by $I$.

Let $\tau_\beta$ be the initial topology on $X_\beta$ with respect to $\family {f_i}_{i \mathop \in I}$.

Let $\phi : \struct{X_\alpha, \tau_\alpha} \to \struct{X_\beta, \tau_\beta}$ be a homeomorphism.

Then:
 * $\tau_\alpha$ is the initial topology on $X_\alpha$ with respect to $\family {f_i \circ \phi : X_\alpha \to Y_i}_{i \mathop \in I}$