We assume a Universe formed by only dark matter and dark energy. The equations of motion that describe the dynamics of the Universe as a whole are the Einstein field equations
\begin{equation}\label{int22}
{R_{\mu \nu }} - \frac{1}{2}R{g_{\mu \nu }} = 8\pi G\left( {{T_{(de)}}_{\mu \nu } + {T_{(dm)}}_{\mu \nu }} \right),
\end{equation}
whereas the conservation equation for each component are
\begin{equation}\label{int23}
\begin{gathered}
{\nabla ^\nu }{T_{(de)}}_{\mu \nu } = {F_\mu }, \\
{\nabla ^\nu }{T_{(dm)}}_{\mu \nu } = - {F_\mu }. \\
\end{gathered}
\end{equation}
where the respective energy momentum tensor for the component $i\;\left( {i = dm,de} \right)$ is
\begin{equation}\label{int24}
{T_{\left( i \right)\mu \nu }} = \left( {{\rho_i} + {p_i}} \right){u_\mu }{u_\nu } - {p_i}{g_{\mu \nu }},
\end{equation}
where ${u_\mu }$ is the velocity of the fluid (the same for each one) and ${\rho_i}$ and ${p_i}$ are respectively the density and pressure of the fluid $i$ measured by an observer with the velocity ${u_\mu }$ . ${F_\mu }$ is the 4-vector of interaction between dark components and its form is not known a priori.
Equations \eqref{int23} can be projected on the time or on the space direction of the comoving observer. We project these equations in a part parallel to the velocity ${u^\mu }$
\begin{equation}\label{int25}
\begin{gathered}
{u^\mu }{\nabla ^\nu }{T_{\left( {dm} \right)\mu \nu }} = - {u^\mu }{F_\mu }, \\
{u^\mu }{\nabla ^\nu }{T_{\left( {de} \right)\mu \nu }} = {u^\mu }{F_\mu }, \\
\end{gathered}
\end{equation}
and in other part orthogonal to the velocity using the projector ${h_{\beta \mu }} = {g_{\beta \mu }} - {u_\beta }{u_\mu }$
\begin{equation} \label{GrindEQ__2_21_}
\begin{array}{l} {\; \; h^{\mu \beta } \nabla ^{\nu } T_{\left(dm\right)\mu \nu } =-h^{\mu \beta } F_{\mu } ,} \\ {\; h^{\mu \beta } \nabla ^{\nu } \nabla ^{\nu } T_{\left(de\right)\mu \nu } =h^{\mu \beta } F_{\mu } }. \end{array}
\end{equation}
Using \eqref{int24} in \eqref{int25} we obtain the energy conservation equations for each dark component
\begin{equation} \label{GrindEQ__2_22_}
\begin{array}{l} {u^{\mu } \nabla_{\mu } \rho_{dm} +\left(\rho_{dm} +p_{dm} \right)\nabla_{\mu } u^{\mu } =u^{\mu } F_{\mu } ,} \\ {u^{\mu } \nabla_{\mu } \rho_{de} +\left(\rho_{de} +p_{de} \right)\nabla_{\mu } u^{\mu } =-u^{\mu } F_{\mu } }. \end{array}
\end{equation}
On the other hand, substitution of \eqref{int24} into \eqref{GrindEQ__2_22_} leads to the Euler equations for each component,
\begin{equation} \label{GrindEQ__2_23_}
\begin{array}{l} {h^{\mu \beta } \nabla_{\mu } p_{dm} +\left(\rho_{dm} +p_{dm} \right)u^{\mu } \nabla_{\mu } u^{\beta } =-h^{\mu \beta } F_{\mu } ,} \\ {h^{\mu \beta } \nabla_{\mu } p_{de} +\left(\rho_{de} +p_{de} \right)u^{\mu } \nabla_{\mu } u^{\beta } =h^{\mu \beta } F_{\mu } } \end{array}.
\end{equation}
We assumed that the background metric is described by the flat FLRW metric . In the comoving coordinates we choose $u^{\mu } =\left(1,0,0,0\right)$. With this choice
\begin{equation} \label{GrindEQ__2_24_}
\begin{array}{l} {\nabla_{\mu } u^{\mu } =3H,} \\ {u^{\mu } \nabla_{\mu } u^{\nu } =0} \end{array}.
\end{equation}
Using the notation $u^{\mu } F_{\mu } =Q(a)$, we transform the equations \eqref{GrindEQ__2_22_} to their final form
\begin{equation} \label{GrindEQ__2_25_}
\begin{array}{l} {\dot{\rho }_{dm} +3H\rho_{dm} =Q,} \\ {\dot{\rho }_{de} +3H(\rho_{de} +p_{de} )=-Q} .\\ {} \end{array}
\end{equation}
The function Q(a) is known as the interaction function, and depends on the scale factor. We note that the equations \eqref{GrindEQ__2_21_} are satisfied identically (taking into account the condition $h^{\mu \nu } F_{\nu } =0$ ) and do not produce any new equations.