smc

First, we notice that $3^0$ is congruent to $1\text{(mod 10)}$, $3^1$ is $3\text{(mod 10)}$, $3^2$ is $9\text{(mod 10)}$, $3^3$ is $7\text{(mod 10)}$, $3^4$ is $1\text{(mod 10)}$, and so on. This turns out to be a cycle repeating every $4$ terms, so $3^{1001}$ is congruent to $3\text{(mod 10)}$. The number $7$ has a similar cycle, going $1,7,9,3,1,...$ Hence we have that $7^{1002}$ is congruent to $9\text{(mod 10)}$. Finally, $13^{1003}$ is congruent to $3^{1003}\equiv 7\text{(mod 10)}$. Thus the required units digit is $3\cdot 9\cdot 7\equiv 9\text{(mod 10)}$, so the answer is $\boxed{9}$.