I am currently going through Conditional Adversarial Nets (CGANs) and the modified objective function of the two-player minimax game is stated as follows: $$\min_G \max_D V(D, G)=\mathbb E_{x\sim p_{data}(x)}[\log D(\textbf x|\textbf y)] + \mathbb E_{z\sim p_z(\textbf z)}[\log(1-D(G(\textbf z | \textbf y)))]$$.

I really don't understand what the conditioning does. G and D are not really probability functions, but rather real functions on samples from a distribution (which are real vectors). So what does the conditioning actually do? If the idea is just to include $y$ as an additional input to the generator neural net, then why not just $D(x, y)$ instead of $D(x|y)$?

I am currently going through Conditional Adversarial Nets (CGANs) and the modified objective function of the two-player minimax game is stated as follows: $$\min_G \max_D V(D, G)=\mathbb E_{x\sim p_{data}(x)}[\log D(\textbf x|\textbf y)] + \mathbb E_{z\sim p_z(\textbf z)}[\log(1-D(G(\textbf z | \textbf y)))].$$

I really don't understand what the conditioning does. G and D are not really probability functions, but rather real functions on samples from a distribution (which are real vectors). So what does the conditioning actually do? If the idea is just to include $y$ as an additional input to the generator neural net, then why not just $D(x, y)$ instead of $D(x|y)$?