Microwave oscillator conditions
The passive terminations \$Γ_g\$ and \$Γ_l\$ must be added so that the input and output resonate at the same frequency of oscillation. For this purpose,  the 2 and 3 conditions are used. In other words, if the oscillator is oscillating at one port then it has to oscillate at the other port too.

there's a proof too for two port oscillator design. Suppose oscillation condition is met at port 1 then from second condition,

\$ \dfrac{1}{Γ_{in}}=Γ_g \$

Now we know, 
\$Γ_{in} = s_{11} + 
           \dfrac{s_{12} \cdot s_{21} \cdot Γ_l}
                 {1-(s_{22} \cdot Γ_l)} \$

substituting Γin formula in the equation, we can easily get, \$ \dfrac{1}{Γ_{out}}=Γ_l\$.

And hence if one port is oscillating then the other port has to oscillate simultaneously. And thats what condition 2 and 3 convey.