$\mu_{xy}$ is the number of agents of type $x$ 'choosing' agents of type $y$.
$\Phi_{xy}$ is the total surplus of a match (between $x$ and $y$). We sometimes decompose this as $\Phi = \alpha + \gamma$.
If agent $x$ and agent $y$ match, they receive utilities
$$U_{xy} + \varepsilon_{y} \ \text{ and } \ V_{xy} + \eta_{x}$$
respectively, where
agents bargain over systematic utilities within feasible set $(U_{xy},V_{xy}) \in \mathcal{F}_{xy}$, which will determine transfers.
$(\varepsilon_y)_{y \in \mathcal{Y} \cup \{ 0 \}}$ and $(\eta_x)_{x \in \mathcal{X} \cup \{ 0 \}}$ are random utility terms with distributions $\mathbf{P}_x$ and $\mathbf{Q}_y$
Choice Problem
For agent of type $x$, the random utility of alternative $y$ is given by $U_{xy} + \varepsilon_y$, where $\varepsilon \sim \mathbf{P}_x$ and $U_{x0} = 0$, hence agent $x$ solves
$$
\max_{y \in \mathcal{Y}} \left\{ U_{xy} + \varepsilon_y, \varepsilon_0 \right\}
$$
Similarly, for an agent of type $y$, the random utility of alternative $x$ is given by $V_{xy} + \eta_x$, where $\eta \sim \mathbf{Q}_y$ and $V_{0y} = 0$, hence agent $y$ solves
$$
\max_{x \in \mathcal{X}} \left\{ V_{xy} + \eta_x, \eta_0 \right\}
$$
Indirect Utility
$G$ is the indirect utility, or $Emax$ operator:
$$
G(U) = \sum_{x \in \mathcal{X}} n_x \mathbb{E}_{\mathbf{P}_x} \left[ \max_{y \in \mathcal{Y}} \left\{ U_{xy} + \varepsilon_y, \varepsilon_0 \right\} \right]
$$
with corresponding operator for $y$:
$$
H(U) = \sum_{y \in \mathcal{Y}} m_y \mathbb{E}_{\mathbf{Q}_y} \left[ \max_{x \in \mathcal{X}} \left\{ V_{xy} + \eta_x, \eta_0 \right\} \right]
$$
By the Williams-Daly-Zachary theorem, the number of agents of type $x$ choosing agents of type $y$ is given by
$$
\mu_{xy} = \frac{\partial G(U)}{\partial U_{xy}}
$$
or
$$
\mu_{xy} = \frac{\partial H(U)}{\partial V_{xy}}
$$
Legendre Transform
If $\mathbf{P}_x$ has a nonvanishing density, the relation between $U$ and $\mu$ is invertible, and one has
$$
\mu_{xy} = \frac{\partial G(U)}{\partial U_{xy}} \ \text{ if, and only if, } \ U_{xy} = \frac{\partial G^*(\mu)}{\partial \mu_{xy}}
$$
where $G^*$ is the Legendre transform:
$$
G^*(\mu) = \max_{U} \left\{ \sum_{xy} \mu_{xy} U_{xy} - G(U) \right\}
$$
Similarly,
$$
\mu_{xy} = \frac{\partial H(V)}{\partial V_{xy}} \ \text{ if, and only if, } \ V_{xy} = \frac{\partial H^*(\mu)}{\partial \mu_{xy}}
$$
where
$$
H^*(\mu) = \max_{V} \left\{ \sum_{xy} \mu_{xy} V_{xy} - H(V) \right\}
$$