Question regarding the factorization of a morphism through an open immersion

Let $K$ be a field, and consider a morphism of locally ringed spaces (or schemes) $f: \operatorname{Spec} K \to Y$. Let $t$ be the unique topological point of $\operatorname{Spec} K$, and suppose that $f(t) \in V$, where $V$ is an affine open subset of $Y$.

My question is: Can $f$ be uniquely factored as $\operatorname{Spec} K \to V \to Y$, where the second map is the natural open immersion?

I understand that this factorization holds trivially at the level of the underlying topological spaces. However, I am not entirely sure if it works out correctly at the level of sheaves of rings (i.e., whether it properly factors as a morphism of locally ringed spaces). Could you explain how the morphisms of sheaves behave in this case and why the factorization is valid?

Any insights would be greatly appreciated. Thank you!