Isomorphism between Algebraic Structures induces Isomorphism between Induced Structures

Theorem
Let $A$ be a set.

Let $\struct {S, \odot}$ and $\struct {T, \otimes}$ be algebraic structures.

Let:
 * $S^A$ denote the set of mappings from $A$ to $S$
 * $T^A$ denote the set of mappings from $A$ to $T$.

Let $\phi$ be an isomorphism from $S$ to $T$.

Let $\chi: S^A \to T^A$ be the mapping defined as:
 * $\forall f \in S^A: \map \chi f = \phi \circ f$

where:
 * $\phi \circ f$ denotes the composition of $\phi$ with $f$.

Then:
 * $\chi$ is an isomorphism from $\struct {S^A, \odot}$ to $\struct {T^A, \otimes}$