Hom functors on a preadditive category are left exact

The underlying Set-valued functor of $\mathcal{C}(A,-)$ is covariantly representable, so [L2] says it preserves every existing small limit and hence every existing finite limit. By [L1], its values and structure maps already lie in $\mathbf{A}\mathbf{b}$, so this is exactly left exactness as an $\mathbf{A}\mathbf{b}$-valued functor.

By [L3], the opposite category ${\mathcal{C}}^{\mathrm{o}\mathrm{p}}$ is again preadditive. The contravariant hom-functor $\mathcal{C}(-,A)$ is the covariantly representable functor ${\mathcal{C}}^{\mathrm{o}\mathrm{p}}(A,-)$, so [L2] applied in ${\mathcal{C}}^{\mathrm{o}\mathrm{p}}$ gives the same finite-limit preservation there. Again [L1] says its values lie in $\mathbf{A}\mathbf{b}$.

Therefore both hom-functors are left exact. This is the representable-functor theorem applied once in $\mathcal{C}$ and once in ${\mathcal{C}}^{\mathrm{o}\mathrm{p}}$, with no separate elementwise exactness computation.