Lov\`{a}sz's hom-counting theorem by inclusion-exclusion principle

Abstract

Let ${\mathcal C}$ be the category of finite graphs. Lov\`{a}sz (1967) shows that if $|\mathrm{Hom}(X,A)|=|\mathrm{Hom}(X,B)|$ holds for any $X$, then $A$ is isomorphic to $B$. Pultr (1973) gives a categorical generalization using a similar argument. Both proofs assume that each object has a finite number of isomorphism classes of subobjects. Generalizations without this assumption are given by Dawar, Jakl, and Reggio (2021) and Regio (2021). Here another generalization without this assumption is given, with a shorter proof. Examples of categories are given, for which our theorem is applicable, but the existing theorems are not.