In 1997, Hofmann and Streicher introduced an explicit construction to lift a Grothendieck universe from the category of sets into the category of set-valued presheaves on a small category. More recently, Awodey presented an elegant functorial analysis of this construction in terms of the categorical nerve, the right adjoint to the functor that takes a presheaf to its category of elements; in particular, the categorical nerve's functorial action on the universal small discrete fibration gives the generic family of the universe's Hofmann-Streicher lifting. Inspired by Awodey's analysis, we define a relative version of Hofmann-Streicher lifting in terms of the right pseudo-adjoint to the 2-functor given by postcomposition with a fibration. Finally, we construct a new 2-bifibration of fibrations in which the opcartesian and cartesian lifts arise from these pseudo-adjunctions.

Ensuring meal safety in hospital settings is of paramount importance, given the specialized dietary needs and the vulnerability of patients. This paper addresses the critical need for accurate food component identification in hospital food service systems to prevent adverse reactions due to allergens or dietary non-compliance. We propose a novel approach to enhance automated meal recognition systems, focusing on the recognition of allergenic food components with priority. We integrate two novel methods within Convolutional Neural Networks: Adaptive Priority Dual Loss Function Strategy and False Negative Weighting with Hyper-parameter Calibration. The results show an improvement in the average recall rate for priority food categories, thereby enhancing meal safety. We compared our method with traditional techniques using benchmark neural network architectures. The findings reveal that our approach significantly improves the recognition rates of priority classes, which is crucial in a hospital setting where accurate food identification can have profound health implications. This research contributes to the field by ofering an innovative solution to a pressing food safety challenge, paving the way for safer and more efficient food service systems in hospitals.

In 1997, Hofmann and Streicher introduced an explicit construction to lift a Grothendieck universe ${\mathcal{U}}$ from Set into the category of Set-valued presheaves on a ${\mathcal{U}}$-small category B. More recently, Awodey presented an elegant functorial analysis of this construction in terms of the categorical nerve, the right adjoint to the functor that takes a presheaf to its category of elements; in particular, the categorical nerve's functorial action on the universal ${\mathcal{U}}$-small discrete fibration gives the generic family of ${\mathcal{U}}$'s Hofmann-Streicher lifting. Inspired by Awodey's analysis, we define a relative version of Hofmann-Streicher lifting in terms of the right pseudo-adjoint to the 2-functor FibA→ FibBgiven by postcomposition with a fibration $p:A \to B$.