Bidirectional case markers in West African languages, including those of the Songhay family, are morphemes inserted between subject and object NPs that would otherwise be adjacent. They therefore specify both that the NP to the left is a subject, and that the NP to the right is an object, and they cannot be bracketed uniquely with either. This is shown by the fact that these morphemes are absent when either subject or object position is (structurally and phonologically) absent, for example due to extraction. This is the only morphological case-marking in the relevant languages. The operation inserting such morphemes must have reference to constituent structure (NP), abstract case (subject, object), and linear adjacency. These data increase the evidence that complex case-marking operations can apply in a centrally located morphology component that has simultaneous access to categorial and linear relations. The idea is questionable that such morphological operations take place at a syntax/PF interface, where syntactic categories are first aligned with prosodic phrases, since actual prosodic (e.g. accentual) bracketings do not always coincide with the bracketings relevant to case morphology. This point is made with data from Tamashek (Berber) nominal prefix alternations, preceding the main section on Songhay case marking.