Order of write and read in Microprocessor interfaced with ADC
To tell the ADC to convert, you need to write to it - that's the WRITE cycle. Once it's done, the data will be available at the ADC's data bus (D0 to D7). You now need to READ this into the controller - that's when the READ signal happens. So yes, it appears that the WRITE cycle happens first, followed by the READ signal.

From the diagram, you can also see an interrupt pin of the MP connected to the ADC, labelled "end of conversion". That's a handy way to let the MP know when to start reading - the MP can stay dormant instead of constantly polling the ADC. That's not too relevant to your question but it will help your understanding. All in all, the MP tells the ADC to start (write cycle), and the ADC interrupts the MP when its done so it can then get the data (read cycle)