Is axiom of constructibility $V = L$ consistent with Tarski–Grothendieck set theory? I wonder what is the relationship between ZF + $V = L$ and Tarski–Grothendieck set theory, because I haven't found any bibliographic references.
If they are compatible, it is possible to introduce $V = L$ using Tarski–Grothendieck as basis instead of ZF.
 A: (I made my answer community wiki because the previous comments cover the important points of the answer.)
You can find the relationship between inaccessible cardinals and Grothendieck universes on Wikipedia (!)

Theorem. The followings are equivalent:

*

*Tarski's axiom A: every set is contained in a Grothendieck universe, and


*There is a proper class of inaccessible cardinals.

Especially, Tarski-Grothendieck set theory $\mathsf{TG}$ and $\mathsf{ZFC}+$"There is a proper class of inaccessibles" are the same theory.
It follows from the fact that every Grothendieck universe is of the form $V_\kappa$ for some inaccessible $\kappa$, where $V_\alpha$ is the $\alpha$th cumulative hierarchy. (See Trevor Wilson's previous answer for the detailed proof.)
Since being inaccessible is downward absolute between $V$ and $L$, we have $$L\models \text{there is a proper class of inaccessible cardinals}$$ if $V$ has a proper class of inaccessibles. (Thank you for Noah Schweber to point it out. Being inaccessible need not be upward absolute.)
It means $\mathsf{TG}+(V=L)$ is consistent if $\mathsf{TG}$ were.
