TITLE: The Freyd-Mitchell Embedding Theorem and projective (injective) objects
QUESTION [8 upvotes]: 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 $R$-$\mathsf{Mod}$.
Wikipedia states

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.)

REPLY [4 votes]: An object $P$ is projective when, given any morphism $f: P \to A$, and any epimorphism $g: B \to A$, there is a unique morphism $h: P \to B$ such that $g \circ h = f$. 
Even if an object $P$ is projective in a full subcategory of R-mod, then, it might not be projective in R-mod itself, because in that case the objects $A$ and $B$ might be outside of the subcategory over which $P$ is projective.