Composition of Borel relations

Question

Let $X,Y,Z$ be Polish spaces, or standard Borel spaces, and let us consider two relations $A \subseteq X \times Y$ and $B \subseteq Y \times Z$ that are Borel sets. Define their composition as $$ C := \{(x,z): \; \exists y \text{ such that }(x,y)\in A \text{ and } (y,z)\in B\}. $$ Is $C$ a Borel subset of $X \times Z$? I can show that it is at least analytic.

Answer 1

Let $X= \{0\}$, $Y = Z = [0,1]$ and $A = X \times Y$. Take $B \subset Y \times Z$ Borel such that its projection to $Z$, $S = \text{proj}_ZB$, is not Borel. Then $C = \{0\}\times S$ is not Borel.