# Is $(\prod R_i)\circ (\prod R_i)$ the same as $\prod R_i\circ R_i$?

For each $i\in I$, $R_i$ is a (binary) relation (on a set $X_i$). Is $(\prod R_i)\circ (\prod R_i)$ the same as $\prod R_i\circ R_i$ as relations on $\prod_{i\in I}X_i$?

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Two elements $(x_i)_{i\in I}$ and $(y_i)_{i\in I}$ of $\prod_{i\in I} X_i$ are related by $\prod_{i\in I} \rho_i$, where each $\rho_i$ is a relation on $X_i$, if $x_i\, \rho_i\, y_i$ for all $i\in I$.
If $\rho$ and $\sigma$ are relations on a set $X$, then $x,y\in X$ are related by $\rho\circ \sigma$ if there is some $z\in X$ such that $x\,\rho\, z\,\sigma\, y$.