Show that the sets $(A^B)^C$ and $A^{B \times C}$ have equal cardinality by constructing an explicit bijection between the two sets.

Question

(Exercise 3.6.6 in Analysis by Tao) Let $A, B, C$ be sets. Show that the sets $(A^B)^C$ and $A^{B \times C}$ have equal cardinality by constructing an explicit bijection between the two sets.

$A^B$ is the collection of all functions from $B$ to $A$. ($\{ g | g: B \to A\}$)

Attempt: We first have that $(A^B)^C = \{F | F : C \to \{g | g : B \to A\}\}$ and $A^{ B\times C} = \{f | f : B \times C \to A\}$. I need to find the bijection $\omega : (A^B)^C \to A^{ B\times C} $.

Could you provide any hint to proceed from here?