Many recursive functions can be defined elegantly as the unique homomorphisms, between two algebras, two coalgebras, or one each, that are induced by some universal property of a distinguished structure. Besides the well-known applications in recursive functional programming, several basic modes of reasoning about scientific models have been demonstrated to admit such an exact meta-theory. Here we explore the potential of coalgebra--algebra homomorphism that are not a priori unique, for capturing more loosely specifying patterns of scientific modelling. We investigate a pair of dual techniques that leverage (co)monadic structure to obtain reasonable genericity even when no universal properties are given. We show the general applicability of the approach by discussing a surprisingly broad collection of instances from real-world modelling practice.