Multivariate categorical with observed data

I’m not sure I understand; in the example above I have two observed categorical variables of size 3. I don’t need the Dirichlet prior to vary across observations, I want to set a single prior per variable.
I thought that the batched dimension of 2 should be on the left in the above example:
observed data dimension (2,3)
prior dimension (2,2)
I feel like I’m missing something very obvious but not sure what it is