Publicly verifiable secret sharing scheme
Here's my understanding:


The input of the hash is $x_i,y_i,a_{1i},a_{2i}$.
The output of the hash is $c$.


The wording is '$c$ is computed as ...'

And later the verifier checks if $c$ matches the Output of the hash function used on the same input variables. If different values were used in the original run, the hash would differ.

So yes, you misunderstood the statement.