Finite-dimensional tensoring preserves projectives in category O

For every $X\in \mathcal{O}$ the tensor-Hom adjunction of [F1] gives a natural isomorphism ${\mathrm{Hom}}_{\mathcal{O}}(E\otimes P,X)\cong {\mathrm{Hom}}_{\mathcal{O}}(P,E^{\ast }\otimes X)$, and $E^{\ast }$ is finite-dimensional $\mathfrak{h}$-semisimple with $E^{\ast }\otimes -$ an exact endofunctor of $\mathcal{O}$.

If $I$ is injective, evaluation and coevaluation for the ordinary contragredient dual $E^{\ast }$ give ${\mathrm{Hom}}_{\mathcal{O}}(X,E\otimes I)\cong {\mathrm{Hom}}_{\mathcal{O}}(E^{\ast }\otimes X,I)$, naturally in $X$. Since $E^{\ast }\otimes -$ is exact by [F1] and $\mathrm{Hom}(-,I)$ is exact by [F3], their composite is exact. Thus $E\otimes I$ is injective.

Since $E\otimes P\in \mathcal{O}$ by [F1], the functor ${\mathrm{Hom}}_{\mathcal{O}}(E\otimes P,-)$ is naturally isomorphic to the composite of the exact functor $X\mapsto E^{\ast }\otimes X$ and the exact functor ${\mathrm{Hom}}_{\mathcal{O}}(P,-)$ of [F2]; composites of exact functors are exact, so ${\mathrm{Hom}}_{\mathcal{O}}(E\otimes P,-)$ is exact and [F2] makes $E\otimes P$ projective in $\mathcal{O}$.

Steps 2.1 and 1.2 prove that finite-dimensional tensoring preserves both projectives and injectives in $\mathcal{O}$.