The space of anisotropic r-contravariant, s-covariant, and \(\alpha \) -homogeneous tensors on a manifold admits a functorial structure where vertical derivatives \(\dot \partial \) and contractions \(\imath _{\mathbb {C}}\) by the Liouville vector field \(\mathbb {C}\) are operators which maintain \(s+\alpha \) constant. In (semi-)Finsler geometry, this structure is transmitted faithfully to connection-type elements yielding the following ladder: geodesic sprays, nonlinear connections, anisotropic connections, and linear (Finslerian) connections. However, it is more loosely transmitted to metric-type ones: Finslerian Lagrangians, Legendre transformations, and anisotropic metrics. We will study this structure in depth and apply it to discuss the recent variational proposals (Einstein-Hilbert, Einstein-Palatini, Einstein-Cartan) for generalizing Einstein equations to the Finsler setting.