This Lecture notes (p. 19) shows that in the problem $$ \begin{aligned} & M C P: & \text { minimize }_{M, y} &-2 \ln (\operatorname{det}(M)) \\ &&\text { s.t. } &\left(\begin{array}{cc}I & M c_{i}-y \\ \left(M c_{i}-y\right)^{T} & 1\end{array}\right) \succeq 0, \quad i=1, \ldots, k, \\ & && M \succ 0 \end{aligned} $$ the matrix inequations are all semidefinite constraints for $M \in \mathbb{S}^{n}_+$ and $y\in \mathbf{R}^n$, since the matrix coefficients are linear functions of the variables $M$ and $y$.
Is $$ f( M, y ) = \left(\begin{array}{cc}I & M c_{i}-y \\ \left(M c_{i}-y\right)^{T} & 1\end{array}\right) $$ the linear function of $M$ and $y$?
From my reading of the notes, I'm assuming you're referring to the line that says "all of the matrix coefficients are linear functions of the variables M and y." He means that the constraints are found by linear functions of M and y. The function f(M,y) is one of them, but note also that the other constraint $M \succ 0$ is also a linear function of $M$ and $y$.