I have the notation for two related models,

`y ~ factor + covariate + (1 + factor | group1) + (1 | group2)`

\begin{aligned}
\operatorname{y}_{i} &\sim N \left(\alpha_{j[i],k[i]} + \beta_{1j[i]}(\operatorname{factor}_{\operatorname{male}}) + \beta_{2}(\operatorname{covariate}), \sigma^2 \right) \\
\left(
\begin{array}{c}
\begin{aligned}
&\alpha_{j} \\
&\beta_{1j}
\end{aligned}
\end{array}
\right)
&\sim N \left(
\left(
\begin{array}{c}
\begin{aligned}
&\mu_{\alpha_{j}} \\
&\mu_{\beta_{1j}}
\end{aligned}
\end{array}
\right)
,
\left(
\begin{array}{cc}
\sigma^2_{\alpha_{j}} & \rho_{\alpha_{j}\beta_{1j}} \\
\rho_{\beta_{1j}\alpha_{j}} & \sigma^2_{\beta_{1j}}
\end{array}
\right)
\right)
\text{, for group1 j = 1,} \dots \text{,J} \\ \alpha_{k} &\sim N \left(\mu_{\alpha_{k}}, \sigma^2_{\alpha_{k}} \right)
\text{, for group2 k = 1,} \dots \text{,K}
\end{aligned}

And 2) `y ~ 0 + factor + covariate + (0 + factor | group1)`

:

\begin{aligned}
\operatorname{y}_{i} &\sim N \left(\beta_{0j[i]}(\operatorname{factor}_{\operatorname{female}}) + \beta_{1j[i]}(\operatorname{factor}_{\operatorname{male}}) + \beta_{2}(\operatorname{covariate}), \sigma^2 \right) \\
\left(
\begin{array}{c}
\begin{aligned}
&\beta_{0j} \\
&\beta_{1j}
\end{aligned}
\end{array}
\right)
&\sim N \left(
\left(
\begin{array}{c}
\begin{aligned}
&\mu_{\beta_{0j}} \\
&\mu_{\beta_{1j}}
\end{aligned}
\end{array}
\right)
,
\left(
\begin{array}{cc}
\sigma^2_{\beta_{0j}} & \rho_{\beta_{0j}\beta_{1j}} \\
\rho_{\beta_{1j}\beta_{0j}} & \sigma^2_{\beta_{1j}}
\end{array}
\right)
\right)
\text{, for group1 j = 1,} \dots \text{,J}
\end{aligned}