t-projective objects as t-flat objects for Hom

Exercise 6.5.5. Let \(C\) be a stable \(\infty \)-category with a t-structure. Setting \(D := C\catop \) and \(E := \Sp \catop \), the hom spectrum functor in \(C\) takes the form \(\hom _C\colon C \times C\catop \to \Sp \catop \).

(1)

Show that this functor satisfies the conditions of Definition 6.5.1: For \(X \in C_{\geq 0}\) and \(Y \in C_{\leq 0}\) we have \(\hom _C(X,Y) \in \Sp _{\leq 0}\).

(2)

Show that an object \(P \in C\) is t-flat with respect to \(\hom _C\) if and only if it is t-projective.