  We equip the basic local crossing bimodules in Ozsv\'ath-Szab\'o's theory of
bordered knot Floer homology with the structure of 1-morphisms of
2-representations, categorifying the $U_q(\mathfrak{gl}(1|1)^+)$-intertwining
property of the corresponding maps between ordinary representations. Besides
yielding a new connection between bordered knot Floer homology and higher
representation theory in line with work of Rouquier and the second author, this
structure gives an algebraic reformulation of a ``compatibility between
summands'' property for Ozsv\'ath-Szab\'o's bimodules that is important when
building their theory up from local crossings to more global tangles and knots.
