Let $X$ and $\{Y_i\}_{i\in I}$ be compact metric spaces (where $I$ an index set of possibly uncountable cardinality). Let $\Gamma_i$ be a compact valued, upper hemicontinuous (UHC) correspondence from $X \rightrightarrows Y_i$ for each $i$.
My question is: Is $\Gamma: X \rightrightarrows \prod_i Y_i$ UHC (with respect to product topology)?, where $\Gamma$ is defined as $\Gamma(x) \mapsto \prod_i \Gamma_i(x)$. I am aware that under more general conditions this is not guaranteed for infinite $I$.
But with compactness of $X$, $Y$'s, and $\Gamma$'s my thinking goes as follows. First define $Y = \prod_i Y_i$. By Tychonoff, $Y$ is compact. Under such conditions we know that UHC implies $Gph(\Gamma_i)$ is closed in $X \times Y_i$. So, define $\hat\Gamma_i: X \rightrightarrows Y$ as
$$ \hat\Gamma_i: x \mapsto \Gamma_i(x) \times \prod_{j\neq i} Y_j $$
Now it seems that $Gph(\hat\Gamma_i)$ is closed in $X \times Y$ (although if my intuition is wrong, this would be my guess for where).
From here, we have that $$Gph(\Gamma) = \bigcap_{i\in I} Gph(\hat\Gamma_i)$$ which, by being the intersection of closed sets, implies that $Gph(\Gamma)$ is closed in $X\times Y$, which, again, under this conditions is equivalent to UHC.
Any help (either affirmation, or pointing out an error) is appreciated. If I am wrong, are there stronger conditions that suffice to ensure UHC of $\Gamma$? What if $I$ is only countably infinite? Thanks.