A morphism of ringed spaces need not be a morphism of locally ringed spaces

Regard $(\{\ast \},A)$ and $(\{\ast \},K)$ as one-point ringed spaces. The assignment $y\mapsto x$ defines a ring homomorphism $A\to K$, so by [F1] it defines a morphism of ringed spaces $(\{\ast \},K)⟶(\{\ast \},A)\mathrm{.}$

The unique maximal ideal of $A$ is $(y)$ by [L1], while the unique maximal ideal of the field $K$ is $0$. Under the stalk map $A\to K$, the element $y$ goes to the nonzero element $x$, hence to a unit of $K$. Therefore the image of $(y)$ is not contained in $0$, so the stalk map is not local. By [F2], the morphism is not a morphism of locally ringed spaces.

This gives a morphism of ringed spaces between locally ringed spaces that is not locally ringed. The statement is false.