Closed immersions are local on the target

Assume $i$ is a closed immersion and fix $j$. The map $i$ restricts to a homeomorphism of $i^{-1}(V_j)$ onto $i(Z)\cap V_j$, which is closed in $V_j$. For $x\in V_j$ with $i(z)=x$ one has $(i_{\ast }{\mathcal{O}}_Z{)}_x={\mathcal{O}}_{Z,z}$ and $(i{\mathrm{\mid }}_{i^{-1}(V_j)}{)}_{\ast }{\mathcal{O}}_{i^{-1}(V_j)}$ has the same stalk at $x$, namely ${\mathcal{O}}_{Z,z}$; the stalk map is the surjection ${\mathcal{O}}_{X,x}\to {\mathcal{O}}_{Z,z}$ of [F1]. Hence ${\mathcal{O}}_{V_j}\to (i{\mathrm{\mid }}_{i^{-1}(V_j)}{)}_{\ast }{\mathcal{O}}_{i^{-1}(V_j)}$ is surjective and the restriction is a closed immersion by [F1].

Conversely assume that every restriction $i^{-1}(V_j)\to V_j$ is a closed immersion, hence injective on points. If $i(z)=i(z^{\mathrm{\prime }})=x$, choose $j$ with $x\in V_j$; then $z,z^{\mathrm{\prime }}\in i^{-1}(V_j)$ and injectivity gives $z=z^{\mathrm{\prime }}$. So $i$ is injective.

For each $j$ the set $i(Z)\cap V_j=i(i^{-1}(V_j))$ is closed in $V_j$, since it is the image of the restriction $i^{-1}(V_j)\to V_j$, a homeomorphism onto a closed subset. A subset of $X$ whose intersection with every member of an open cover is closed is closed, so $i(Z)$ is closed in $X$.

Let $C\subseteq Z$ be closed. Then $C\cap i^{-1}(V_j)$ is closed in $i^{-1}(V_j)$, so its image $i(C)\cap V_j=i(C\cap i^{-1}(V_j))$ is closed in $i(Z)\cap V_j$ for every $j$, using the first direction of [F1] for each restriction. Hence $i(C)$ is closed in the closed subset $i(Z)$, and $i:Z\to i(Z)$ is a homeomorphism.

Finally ${\mathcal{O}}_X\to i_{\ast }{\mathcal{O}}_Z$ is surjective, because surjectivity of a morphism of sheaves is checked on stalks and at $x\in V_j$ the stalk map is the stalk at $x$ of the surjective map ${\mathcal{O}}_{V_j}\to (i{\mathrm{\mid }}_{i^{-1}(V_j)}{)}_{\ast }{\mathcal{O}}_{i^{-1}(V_j)}$ of step 1.2.

Steps 1.2, 2.1, 2.2 and 2.3 exhibit $i$ as a homeomorphism onto the closed subset $i(Z)$ with surjective structure map, so $i$ is a closed immersion by [F1]; the forward direction is step 1.1. ∎