Probability notation q(y) and q(Y) and its implication to vector functions

The function in question is (from Appendix B, Proof of proposition 2.1 from Posterior Regularization for Structured Latent Variable Models):

$$q(\textbf{Z}) = \frac{p_{\theta}(\textbf{Z}|\textbf{X})exp(\lambda^T \cdot \Phi(\textbf{Z}|\textbf{X}))}{H}$$

The $q(\textbf{Z})$ is a probability distribution of latent variable $z$, such that $\textbf{Z} \in \mathbb{R}^{N \times 1}$, $\textbf{X}$ is a vector of $N$ datapoints such that $\textbf{X} \in \mathbb{R}^{N \times 2}$. The $\lambda$ is a dual variable, such that $\lambda \in \mathbb{R}^{N \times 1}$ and is a vector function $\Phi(\textbf{Z}|\textbf{X})$. $H$ is a constant that normalizes the distribution to have a sum of $1$.

The question is, how to evaluate $q()$ for a single variable $z$, rather than a vector $\textbf{Z}$ ?

Particular example of confusion: For instance, for particular $z_i$, which is related to datapoint $x_j$, is the term in exponential this $exp(\lambda^T \cdot \Phi(\textbf{Z}|\textbf{X}) )$ or this $exp(\lambda^T_i \cdot \Phi(z_i|x_j) )$?

In other words, is a single $z$ weighted through the exponential term obtained from all other $\textbf{Z}$, or just the particular $z_i$?

If its the former case, do I need to explicitly define $\Phi(z_k|x_j) = 0$, when $i \ne k$, because otherwise it seems undefined as a vector function for the elements $\Phi()$ not defined for $z_i$.

General question: How to intuitively understand whether my distribution/function is a function of only particular variable or a vector of variables?