From reading the literature, it seems that if we have a general conic optimization problem given by,
\begin{equation} \begin{aligned} & \underset{x}{\text{min}} & & c^T x \\ & \text{s.t.} & & Ax = b \\ & & & Gx + s = h \\ & & & s \succeq_K 0 \end{aligned} \end{equation}
and its dual problem given by,
\begin{equation} \begin{aligned} & \underset{y,z}{\text{max}} & & -b^T y - h^T z \\ & \text{s.t.} & & A^T y + G^T z + c = 0 \\ & & & z \succeq_{K^{*}} 0 \\ \end{aligned} \end{equation} where $K=K^*$ (self-dual), then the KKT conditions are given by,
\begin{equation} \begin{aligned} A^T y + G^T z + c = 0 \\ Ax - b = 0 \\ Gx + s - h = 0 \\ s^T \circ z = 0 \\ s,z \succeq_K 0 \\ \end{aligned} \end{equation}
Since this topic has such a steep learning curve, I wanted to clarify a few things to see if I'm on the right track. Would the first KKT equation imply dual equality constraint feasibility, the second equation primal equality constraint feasibility, and the third equation primal inequality constraint feasibility? I can see that the fifth equation is simply enforced from the original primal and dual problems, but where does $s^T z = 0$ come from? Is this just saying that the duality/complementarity gap between the primal and dual problem must be zero? If so this would imply that $$s^T \circ z = c^T x - (- b^T y - h^T z) = c^T x + b^T y + h^T z = 0$$ but I don't quite know how to get that. Any clarification would be much appreciated.