The Cook-Levin reduction map is computable in polynomial time

On an input $w$ of length $n$, [L3] gives a tableau side length $T+1$ with $T\le p(n)$. A deterministic transducer can loop over all cells, all alphabet symbols, and all local windows in lexicographic order and print the corresponding clauses of ${\psi }_{N,w}$.

By [L1] and [L2], the number of printed variables and clauses is polynomial in $T$, hence polynomial in $n$ because $T\le p(n)$. Each loop iteration performs only constant-size bookkeeping for the fixed machine $N$.

Therefore the transducer from step 1.1 halts after polynomially many steps and outputs ${\psi }_{N,w}$. So the Cook-Levin map $w\mapsto {\psi }_{N,w}$ is polynomial-time computable.