The proposition says that
If $(R,m,k)$ is a Cohen-Macaulay (CM) local ring of dimension $d$ and $C$ is a maximal Cohen-Macaulay (MCM) $R$-Module, then
a) Suppose $M$ is a MCM $R$-module with $Ext^j_R(M,C)=0$ for all $j>0$. Then $Hom_R(M,C)$ is MCM $R$-module, and for any $R$-sequence $X$ we have $Hom_R(M,C)\otimes R/XR \cong Hom_{R/XR}(M/XM,C/XC)$.
b) Assume in addition that $C$ has finite injective dimension and $M$ is a CM $R$-module of dimension $t$. Then
i) $Ext^j_R(M,C)=0$ for all $j\neq d-t$.
ii) $Ext^{d-t}_R(M,C)$ is a CM module of dimension $t$.
I could prove the isomorphism in the statement a) but was unable to understand how to verify that the $Hom$ set is MCM.
For part b) i) the book refers to Theorem 1.2.10 (e) which says that if $I$ is an ideal of $R$ and $M, N$ are finite $R$-modules with $Supp(N)=V(I)$ then $grade(I,M)=inf\{i: Ext^i(N,M)\neq 0\}$. The author claims that statement b) i) follows from the above statement for $j<d-t$, which I am not able to figure out.
I am not equipped with much familiarity with these things so a detailed answer will be helpful to me. I am also looking for a lucid reference from where I can learn the proof of the statements.
I really appreciate any help you can provide.