complexity theory – Time constructible function T in equivalence of Turing Machines

I’m reading Computational Complexity by Arora and Barak and I had a doubt regarding a statement made about the equivalence of Turing machines:

For every $fcolon {0, 1}^∗ → {0, 1}$ and time-constructible $Tcolon mathbb N → mathbb N$, if $f$ is computable in time $T (n)$
by a Turing machine $M$ using alphabet $Gamma$ then it is computable in time $4log|Gamma|T (n)$ by a Turing machine $tilde{M}$ using the
alphabet ${0, 1, square, B}$.

Specifically, what if any role $T$ being time constructible has here?

The proof (sketch) encodes $Gamma$ in binary, and uses a larger register to simulate $M$‘s transition function, and also a counter. But I don’t think $T$ being time constructible comes into play. Could someone confirm or tell me if I’m missing something?