  A classical result by Lov\'asz asserts that two graphs $G$ and $H$ are
isomorphic if and only if they have the same left profile, that is, for every
graph $F$, the number of homomorphisms from $F$ to $G$ coincides with the
number of homomorphisms from $F$ to $H$. Dvor{\'{a}}k and later on Dell, Grohe,
and Rattan showed that restrictions of the left profile to a class of graphs
can capture several different relaxations of isomorphism, including equivalence
in counting logics with a fixed number of variables (which contains fractional
isomorphism as a special case) and co-spectrality (i.e., two graphs having the
same characteristic polynomial). On the other side, a result by Chaudhuri and
Vardi asserts that isomorphism is also captured by the right profile, that is,
two graphs $G$ and $H$ are isomorphic if and only if for every graph $F$, the
number of homomorphisms from $G$ to $F$ coincides with the number of
homomorphisms from $H$ to $F$. In this paper, we embark on a study of the
restrictions of the right profile by investigating relaxations of isomorphism
that can or cannot be captured by restricting the right profile to a fixed
class of graphs. Our results unveil striking differences between the expressive
power of the left profile and the right profile. We show that fractional
isomorphism, equivalence in counting logics with a fixed number of variables,
and co-spectrality cannot be captured by restricting the right profile to a
class of graphs. In the opposite direction, we show that chromatic equivalence
cannot be captured by restricting the left profile to a class of graphs, while,
clearly, it can be captured by restricting the right profile to the class of
all cliques.
