Adjointness properties for relative extensions of disk and sphere chain complexes

We study the subgroup $\mathcal{E}xt^i_{\mathcal{C}}(\mathcal{F}; C, D)$ of $\mathcal{E}xt^i_{\mathcal{C}}(C,D)$ 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 $\mathcal{C} \longrightarrow \textbf{Ch}(\mathcal{C})$, given by a collection of natural isomorphisms which generalize the corresponding adjointness properties proven by J. Gillespie for $\mathcal{E}xt^i(-,-)$.