We show that a combinatorial recipe to define the Bates--Pask (proper) outsplit is equivalent to a morphism of quivers satisfying some distinguished categorical properties. As a formal consequence, we reveal the existence of a canonical convolution type hypercategory structure on proper Bates--Pask out-split moves on quivers which is provided with a Leavitt path algebra functor to the groupoid of graded *-algebras. Here, closedness of out-splits under a convolution type composition stems from the algebraic property of associative unital pasting of Cartesian squares for target maps of quivers, and a coalgebraic property of coassociative counital decomposition of split squares for source maps. It implies also that Bates--Pask (proper) out-splits are stable under (proper) retracts.