Compatible open pieces of ringed or locally ringed spaces glue

Form the disjoint union ${\coprod }_i{X}_i$ and impose the equivalence relation generated by $x_i\sim {\phi }_{ij}(x_i)$ whenever $x_i\in U_{ij}$, using the underlying homeomorphisms of the isomorphisms ${\phi }_{ij}$. The identity and cocycle hypotheses make this an equivalence relation. Let $X$ be the quotient space. The image $V_i$ of each $X_i$ in $X$ is open, the $V_i$ cover $X$, and the quotient map $q_i:X_i\to V_i$ is a homeomorphism because identifications occur only along open subsets via homeomorphisms.

Transport each structure sheaf ${\mathcal{O}}_i$ to the open subset $V_i$ via $q_i$, obtaining a sheaf of rings on $V_i$. The ringed-space isomorphisms on the overlaps induce isomorphisms of these sheaves on $V_i\cap V_j$, and the cocycle condition is exactly the compatibility required by [L1]. Hence [L1] glues the local structure sheaves to a sheaf of rings ${\mathcal{O}}_X$ on $X$, making each $q_i$ an isomorphism of ringed spaces $(X_i,{\mathcal{O}}_i)\cong (V_i,{\mathcal{O}}_X{\mathrm{\mid }}_{V_i})$.

If the input pieces are locally ringed spaces, let $x\in X$ and choose $i$ with $x\in V_i$. Because $q_i$ is a homeomorphism onto an open neighbourhood of $x$, the stalk ${\mathcal{O}}_{X,x}$ identifies with the stalk of ${\mathcal{O}}_i$ at the corresponding point of $X_i$. That stalk is local by [F1], so every stalk of ${\mathcal{O}}_X$ is local. Thus $(X,{\mathcal{O}}_X)$ is locally ringed.

Any other glued ringed or locally ringed space has the same quotient-topology description as in step 1.1 and the same locally compatible structure sheaf. By the uniqueness clause of [L1], it is uniquely isomorphic to $(X,{\mathcal{O}}_X)$. This proves existence and uniqueness in both the ringed and locally ringed settings.