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Let ${\mathcal C}$ be the category of finite graphs. Lov\`{a}sz (1967) showsthat 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 asimilar argument. Both proofs assume that each object has a finite number ofisomorphism classes of subobjects. Generalizations without this assumption aregiven by Dawar, Jakl, and Reggio (2021) and Regio (2021). Here anothergeneralization without this assumption is given, with a shorter proof. Examplesof categories are given, for which our theorem is applicable, but the existingtheorems are not.

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