Given a small abelian category $\mathcal{A}$, the Freyd-Mitchell Embedding Theorem gives me a fully faithful exact functor $F:\mathcal{A}\rightarrow R$-$\mathsf{Mod}$, for some unital ring $R$, so that, in particular, $\mathcal{A}$ is equivalent to a full subcategory of $R$-$\mathsf{Mod}$.
However, projective and injective objects in $\mathcal{A}$ do not necessarily correspond to projective and injective $R$-modules.
Given that the definition of projective objects is entirely categorical in nature, how could it be that the notions of projective in each category do not correspond to each other in equivalent categories? Is this because the projective objects in a full subcategory of $R\mathsf{Mod}$ are sometimes different than the projective objects in all of $R$-$\mathsf{Mod}$?
(Of course, you can everywhere replace "projective" with "injective" in the above.)