Concatenation of two Turing machines
Given two Turing machines $M_1$ and $M_2$, assume that their state sets are disjoint, i.e, $Q_1 \cap Q_2 = \emptyset$. Define $\Sigma_3 = \Sigma_1 \cup \Sigma_2, Q_3=Q_1\cup Q_2, F_3=F_2$. 

Modify $M_1$ so that it prints its output starting from the first (leftmost) tape cell, then move the tape head to the leftmost cell and enter the $M_2$'s start state. This makes sure that $M_2$'s input is the $M_1$'s output. You may also need to add new states to $Q_3$ .