Naturality of the connecting morphism

Regard each vertical arrow of the given ladder as an object of the arrow category ${\mathcal{A}}^{\to }$. Because kernels, cokernels, pullbacks, and pushouts in ${\mathcal{A}}^{\to }$ are computed componentwise, the entire ladder is again a Mac Lane snake diagram in ${\mathcal{A}}^{\to }$.

Applying [L2] in ${\mathcal{A}}^{\to }$ produces a connecting morphism between the arrow objects $\ker (\gamma )\to \mathrm{coker}(\alpha )\mathrm{.}$ Read componentwise in $\mathcal{A}$, that morphism is exactly the pair consisting of the two ordinary connecting morphisms together with the comparison square between them.

The defining square for the arrow-category connecting morphism commutes by construction, and uniqueness in [L2] forces that componentwise square to be the naturality square for the two ordinary connecting morphisms. Therefore the connecting morphism is natural.