Homeomorphic Topology of Initial Topology is Initial Topology

Theorem
Let $\struct{X_1, \tau_1}, \struct{X_2, \tau_2}$ be topological spaces.

Let $\phi : \struct{X_1, \tau_1} \to \struct{X_2, \tau_2}$ be a homeomorphism.

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 $\family {f_i: X_2 \to Y_i}_{i \mathop \in I}$ be an indexed family of mappings indexed by $I$.

Let $\tau_2$ be the initial topology on $X_2$ with respect to $\family {f_i}_{i \mathop \in I}$.

Then:
 * $\tau_1$ is the initial topology on $X_1$ with respect to $\family {\phi \circ f_i : X_1 \to Y_i}_{i \mathop \in I}$