  In this paper, meant as a companion to arXiv:2006.04458, we consider a class
of non-integrable $2D$ Ising models in cylindrical domains, and we discuss two
key aspects of the multiscale construction of their scaling limit. In
particular, we provide a detailed derivation of the Grassmann representation of
the model, including a self-contained presentation of the exact solution of the
nearest neighbor model in the cylinder. Moreover, we prove precise asymptotic
estimates of the fermionic Green's function in the cylinder, required for the
multiscale analysis of the model. We also review the multiscale construction of
the effective potentials in the infinite volume limit, in a form suitable for
the generalization to finite cylinders. Compared to previous works, we
introduce a few important simplifications in the localization procedure and in
the iterative bounds on the kernels of the effective potentials, which are
crucial for the adaptation of the construction to domains with boundaries.
