Let $C$ and $D$ be categories such that $C\simeq D$. Also, let $C'$ and $D'$ be categories such that $C'\simeq D'$. Then $C\times C'\simeq D\times D'$.
My attempt:
Since $C\simeq D$, there exists functors $F:C\to D$ and $G:D\to C$ along with natural isomorphisms $\alpha:id_C\simeq GF$ and $\beta:FG\simeq id_D$. Since $C'\simeq D'$, there exists functors $F':C'\to D'$ and $G':D'\to C'$ along with natural isomorphisms $\alpha':id_{C'}\simeq G'F'$ and $\beta':F'G'\simeq id_{D'}$. We know $F\times F':C\times C'\to D\times D'$ defined where $(F\times F')(c,c')=(Fc,F'c')$ for all $(c,c')\in C\times C'$ is a functor and $G\times G':D\times D'\to C\times C'$ defined where $(G\times G')(d,d')=(Gd,G'd')$ for all $(d,d')\in D\times D'$ is a functor. Let $(d,d')\in D\times D'$. Then we have that $(F\times F')((G\times G')(d,d'))=(F\times F')(Gd,G'd')=(FGd,F'G'd')=(FG\times F'G')(d,d')$. This must mean $(F\times F')(G\times G')=FG\times F'G'$. Similarly $(G\times G')(F\times F')=GF\times G'F'$. Note that there exists natural isomorphisms $\alpha\times \alpha':id_C\times id_{C'}\simeq GF\times G'F'$ and $\beta\times \beta':FG\times F'G'\simeq id_D\times id_{D'}$. Another way to write these natural isomorphisms is $\alpha\times \alpha':id_{C\times C'}\simeq (G\times G')(F\times F') $ and $ \beta\times \beta':(F\times F')(G\times G')\simeq id_{D\times D'}$. So, $C\times C'\simeq D\times D'$.
How is it? If it is not good, how do I fix it?