Given a gluing datum of locally ringed spaces, i.e.
- a family $X_i, i \in I$ of locally ringed spaces,
- for all $i,j \in I$ an open subset $X_{ij} \subseteq X_i$,
- for all $i,j$, an isomorphism of locally ringed spaces $\phi_{ij}\colon X_{ij}\to X_{ji}$,
such that $U_{ii} = U_i, \phi_{ii} = 1_{U_i}$ and $\phi_{ik} = \phi_{jk}\circ \phi_{ij}$ on $X_{ij}\cap X_{ik}$,
one can glue $X_i$ along $\phi_{ij}$ to obtain a new locally ringed space $X$ (together with canonical open immersions $\psi_i\colon X_i\to X$) satisfying a certain universal property. I'm trying to find out what this property is exactly.
Gortz and Wedhorn (the book Algebraic Geometry I: Schemes: With Examples and Exercises) state it as follows: if $Y$ is a locally ringed space and for all $i \in I \ \ \psi_i\colon X_i\to T$ a morphism of locally ringed spaces which induce an isomorphism of $X_i$ with with an open subspace of $T$, such that $\xi_j\circ \phi_{ij} = \xi_i$ on $U_{ij}$ for all $i,j \in I$, then there exists a unique morphism $\xi\colon X\to T$ with $\xi\circ \psi_i = \xi_i$ for all $i \in I$.
According to the book, the claim follows from the existence of the gluing of locally ringed spaces and the gluing lemma for morphisms (which states that we can glue morphisms $f_i\colon U_i\to X$ of locally ringed spaces where $U_i \subseteq Y$ to a unique morphism $f\colon Y\to X$).
What I don't see is why it's necessary that $\xi_i$ are open immersions ($\psi_i$ necessarily are though). Unless I'm mistaken (which is very possible), the deduction of the universal property of the gluing from the gluing lemma for morphisms can be done without this extra assumption. Stacks doesn't seem to include this line as well.