The structure morphism of a projective variety induces a morphism of $k$-algeras

Question

Suppose that $k$ is an algebraically closed field and that $X=\textrm{Proj}{\frac{k[T_1,\ldots,T_n]}{(f_1,\ldots,f_n)}}$ is a projective variety with a structural morphism $p:X\rightarrow\textrm{Spec} k$. Does the morphism $p$ induce an univocally determined morphism of $k$-algebras $k\longrightarrow\frac{k[T_1,\ldots,T_n]}{(f_1,\ldots,f_n)}\;\textrm{?}$

In the case of affine varieties this is true because $\textrm{Spec}()$ is a functor, but I suspect that the condition holds also for projective varieties, without invoking any "glueing" argument.

Thanks in advance.

Answer 1

$k$ is the initial $k$-algebra.