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