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Aims. Our objectives are to map the filamentary network around the Fornax-Eridanus complex and probe the influence of the local environment on galaxy morphology. Methods. We employed the novel machine-learning tool, named, 1-Dimensional, Recovery, Extraction, and Analysis of Manifolds (1-DREAM) to detect and model filaments around the Fornax cluster. We then used the morphology-density relation of galaxies to examine the variation in the galaxies’ morphology with respect to their distance from the central axis of the detected filaments. Results. We detected 27 filaments that vary in length and galaxy-number density around the Fornax-Eridanus complex. We find that 81% of galaxies in our catalogue belong to filaments and 19% of galaxies are located outside filaments. The filaments around the Fornax-Eridanus complex showcase a variety of environments: some filaments encompass groups and clusters, while others are only inhabited by galaxies in pristine filamentary environments. In this context, we reveal a well-known structure, namely: the Fornax Wall, which passes through the Dorado group, Fornax cluster, and Eridanus supergroup. With regard to the morphology of galaxies, we find that early-type galaxies (ETGs) populate high-density filaments and high-density regions of the Fornax Wall. Furthermore, the fraction of the ETG-population decreases as the distance to the central axis of the filament increases. The fraction of late-type galaxies (LTGs; 8%) is lower than that of ETGs (12%) at 0.5 Mpc/ h from the filament spine. Of the total galaxy population in filaments around the Fornax-Eridanus complex, ∼7% are ETGs and ∼24% are LTGs located in pristine environments of filaments, while ∼27% are ETGs and ∼42% are LTGs in groups and clusters within filaments. Among the galaxies in the filamentary network around the Fornax-Eridanus complex, 44% of them belong to the Fornax Wall. Conclusions. This study reveals the cosmic web around the Fornax cluster, which exhibits a variety of filamentary environments. With this, our research asserts that filamentary environments are heterogeneous in nature. When investigating the role of the environment on galaxy morphology, it is essential to consider both the local number-density and a galaxy’s proximity to the filament spine (i.e. the filament core). Within this framework, we ascribe the observed morphological segregation in the Fornax Wall to the pre-processing of galaxies among groups embedded in it.