Homomorphisms are indeed a good basis for counting: Three fixed-template dichotomy theorems, for the price of one.

Many natural combinatorial quantities can be expressed by counting the number of homomorphisms to a fixed relational structure. For example, the number of 3-colorings of an undirected graph $G$ is equal to the number of homomorphisms from $G$ to the $3$-clique. In this setup, the structure receiving the homomorphisms is often referred to as a template; we use the term template function to refer to a function, from structures to natural numbers, that is definable as the number of homomorphisms to a fixed template. There is a literature that studies the complexity of template functions. The present work is concerned with relating template functions to the problems of counting, with respect to various fixed templates, the number of two particular types of homomorphisms: surjective homomorphisms and what we term condensations. In this article, we explain how any problem of counting surjective homomorphisms to a fixed template is polynomial-time equivalent to computing a linear combination of template functions; we also show this for any problem of counting condensations to a fixed template. Via a theorem that characterizes the complexity of computing such a linear combination, we show how a known dichotomy for template functions can be used to infer a dichotomy for counting surjective homomorphisms on fixed templates, and likewise a dichotomy for counting condensations on fixed templates. Our study is strongly inspired by, based on, and can be viewed as a dual of the graph motif framework of Curticapean, Dell, and Marx (STOC 2017); that framework is in turn based on work of Lov\'asz (2012).