Gorenstein projective objects in functor categories

Let $k$ be a commutative ring, let $\mathcal{C}$ be a small, $k$-linear, Hom-finite, locally bounded category, and let $\mathcal{B}$ be a $k$-linear abelian category. We construct a Frobenius exact subcategory $\mathcal{G}\mathcal{P}({\mathcal{G}}_P\mathrm{proj}({\mathcal{B}}^{\mathcal{C}}))$ of the functor category ${\mathcal{B}}^{\mathcal{C}}$, and we show that it is a subcategory of the Gorenstein projective objects $\mathcal{G}\mathcal{P}({\mathcal{B}}^{\mathcal{C}})$ in ${\mathcal{B}}^{\mathcal{C}}$. Furthermore, we obtain criteria for when $\mathcal{G}\mathcal{P}({\mathcal{G}}_P\mathrm{proj}({\mathcal{B}}^{\mathcal{C}}))=\mathcal{G}\mathcal{P}({\mathcal{B}}^{\mathcal{C}})$. We show in examples that this can be used to compute $\mathcal{G}\mathcal{P}({\mathcal{B}}^{\mathcal{C}})$ explicitly.