In a multi-tape Turing machine, suppose there are $k$ tapes.

At any step, the machine reads:

$k$ tape symbols

So, input to transition function is:

$current\ state = Q$

$k\ tape\ symbols = \Gamma^k$

Hence input side is:

$Q \times \Gamma^k$

After transition, the machine decides:

new state $Q$

new symbols to write on $k$ tapes, that is $\Gamma^k$

movement of $k$ tape heads, that is ${L,R,S}^k$

Hence output side is:

$Q \times \Gamma^k \times {L,R,S}^k$

Therefore,

$\delta: Q \times \Gamma^k \to Q \times \Gamma^k \times {L,R,S}^k$