Adjointness properties for relative extensions of disk and sphere chain complexes

We study the subgroup $\mathcal{E}x{t}_{\mathcal{C}}^i(\mathcal{F};C,D)$ of $\mathcal{E}x{t}_{\mathcal{C}}^i(C,D)$ formed by those $i$-extensions of $C$ by $D$ in an Abelian category $\mathcal{C}$ which are ${\rm Hom}{$\mathcal{C}$}($\mathcal{F}$,-)$-exact, and present a Baer-like description of this subgroup in terms of certain right derived functors of ${\rm Hom}{$\mathcal{C}$}(-,-)$. We also study adjointness properties of these subgroups and the disk and sphere chain complex functors $$\mathcal{C}$ \longrightarrow $\mathbf{\text{Ch}}$($\mathcal{C}$)$, given by a collection of natural isomorphisms which generalize the corresponding adjointness properties proven by J. Gillespie for $\mathcal{E}x{t}^i(-,-)$.