  We develop a category-theoretic criterion for determining the equivalence of
causal models having different but homomorphic directed acyclic graphs over
discrete variables. Following Jacobs et al. (2019), we define a causal model as
a probabilistic interpretation of a causal string diagram, i.e., a functor from
the ``syntactic'' category $\textsf{Syn}_G$ of graph $G$ to the category
$\textsf{Stoch}$ of finite sets and stochastic matrices. The equivalence of
causal models is then defined in terms of a natural transformation or
isomorphism between two such functors, which we call a $\Phi$-abstraction and
$\Phi$-equivalence, respectively. It is shown that when one model is a
$\Phi$-abstraction of another, the intervention calculus of the former can be
consistently translated into that of the latter. We also identify the condition
under which a model accommodates a $\Phi$-abstraction, when transformations are
deterministic.
