Here is a definition of holomorphic convexity taken from the notes of Eyssidieux:


**Defintion.** A complex analytic space $S$ is holomorphically convex if there is a proper holomorphic morphism $\pi: S\to T$ with $\pi\_\*O\_S=O\_T$ such that $T$ is a Stein space. $T$ is then called *Cartan-Remmert* reduction of $S$.


**Questions**. 1) Is this correct that Cartan-Remmert reduction is unique is if exists? 


2) Do I understand correctly, that (assuming properness of $\pi$) $\pi\_\*O\_S=O\_T$ just means that $\pi$ is a surjective map and all its fibers are connected?



A basic question on the definition of Cartan-Remmert reduction  and holomorphic convexity
A full statement of the Cartan-Remmert reduction includes also a universal property which should answer your question (you will get uniqueness up to a unique isomorphism ):
in the Encyclopedia of Math. Sciences (several complex variables, vol. 7) you will find the following:


Let $X$ be a holomorphically convex space. Then there exists a Stein space $Y$ and a proper surjective holomorphic map $\phi:X \rightarrow Y$ with the following properties:


1) $\phi$ has connected fibers,


2) $\phi\_{\star}O\_{X}=O\_Y$,


3) the canonical map $O\_{Y}(Y) \rightarrow O\_{X}(X)$ is an isomorphism,


4)(universal property) if $\sigma:X \rightarrow Z$ is a holomorphic map into a Stein space $Z$, then there exists a uniquely determined holomorphic map $\tau:Y \rightarrow Z$ such that the diagram $\phi:X \rightarrow Y$, $\tau: Y \rightarrow Z$, $\sigma: X \rightarrow Z$ commutes.


Remark: the diagram mentioned above should be seen as a triangle (I couldn't type a commutative diagram...)


