@dogodel there is, the next step is mere correspondence instead of unidirectional arrow. https://ncatlab.org/nlab/show/span
Lax definitions replace isomorphism requirements by one-directional morphism requirements.
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@dogodel The abstract way to generalize this is by using various notions of fibration, though https://ncatlab.org/nlab/show/Cartesian+fibration#cartesian_fibrations_over_one_morphism_and_adjunctions …
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