Why $Z_1\to Z_2$ is necessarily unique? 
Why the isomorphism $Z_1\to Z_2$ is necessarily unique?
 A: This follows from the fact that closed immersions are monomorphisms in the category of schemes.
To see this, let $f: Z \rightarrow X$ be a closed immersion of schemes.  Let $\phi_1, \phi_2$ be morphisms of schemes $Y \rightarrow Z$ such that $f \circ \phi_1= f \circ \phi_2$.  We want to show that $\phi_1 = \phi_2$.
On the level of topological spaces, $f$ is a homeomorphism onto a closed subset of $X$, and in particular is injective.  So we can conclude that $\phi_1 = \phi_2$ as continuous maps $Y \rightarrow Z$.  Let's call this map $\phi$.
We also need to show that the morphisms of sheaves $\phi_1^{\#}, \phi_2^{\#}: \mathcal O_Z \rightarrow \phi_{\ast} \mathcal O_Y$ are equal. We have
$$f_{\ast}(\phi_1^{\#}) \circ f^{\#} = (f \circ \phi_1)^{\#} = (f \circ \phi_2)^{\#} =  f_{\ast}(\phi_2^{\#}) \circ f^{\#}$$
Since $f$ is a closed immersion, $f^{\#}: \mathcal O_X \rightarrow f_{\ast}(\mathcal O_Z)$ is a surjective morphism of sheaves on $X$.  Such morphisms are epimorphisms in the category of sheaves, so we can conclude that $f_{\ast}(\phi_1^{\#}) = f_{\ast}(\phi_2^{\#})$.  Using the definition of $f_{\ast}$ and the fact that every open set in $Z$ is the preimage under $f$ of an open set in $X$, it follows easily that $\phi_1^{\#} = \phi_2^{\#}$.
